In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, an eigenvector () or characteristic vector of a
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
is a nonzero
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
that changes at most by a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by
, is the factor by which the eigenvector is scaled.
Geometrically, an eigenvector, corresponding to a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
nonzero eigenvalue, points in a direction in which it is
stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, the eigenvector is not rotated.
Formal definition
If is a linear transformation from a vector space over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
into itself and is a
nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as
where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root associated with .
There is a direct correspondence between ''n''-by-''n''
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
and linear transformations from an
''n''-dimensional vector space into itself, given any
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, or the language of linear transformations.
If is finite-dimensional, the above equation is equivalent to
where is the matrix representation of and is the
coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
of .
Overview
Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix ''
eigen-'' is adopted from the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
word ''
eigen Eigen may refer to:
* Eigen (C++ library), computer programming library for matrix and linear algebra operations
* Eigen Technologies, the Document AI software company
* Eigen, Schwyz, settlement in the municipality of Alpthal in the canton of Schw ...
'' (
cognate
In historical linguistics, cognates or lexical cognates are sets of words in different languages that have been inherited in direct descent from an etymology, etymological ancestor in a proto-language, common parent language. Because language c ...
with the
English
English usually refers to:
* English language
* English people
English may also refer to:
Peoples, culture, and language
* ''English'', an adjective for something of, from, or related to England
** English national ide ...
word ''
own
Ownership is the state or fact of legal possession and control over property, which may be any asset, tangible or intangible. Ownership can involve multiple rights, collectively referred to as title, which may be separated and held by different ...
'') for 'proper', 'characteristic', 'own'.
Originally used to study
principal axes of the rotational motion of
rigid bodies
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
, eigenvalues and eigenvectors have a wide range of applications, for example in
stability analysis,
vibration analysis
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, such ...
,
atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
s,
facial recognition, and
matrix diagonalization
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
.
In essence, an eigenvector v of a linear transformation ''T'' is a nonzero vector that, when ''T'' is applied to it, does not change direction. Applying ''T'' to the eigenvector only scales the eigenvector by the scalar value ''λ'', called an eigenvalue. This condition can be written as the equation
referred to as the eigenvalue equation or eigenequation. In general, ''λ'' may be any
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
. For example, ''λ'' may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
.
![Mona Lisa eigenvector grid](https://upload.wikimedia.org/wikipedia/commons/3/3c/Mona_Lisa_eigenvector_grid.png)
The
Mona Lisa
The ''Mona Lisa'' ( ; it, Gioconda or ; french: Joconde ) is a half-length portrait painting by Italian artist Leonardo da Vinci. Considered an archetypal masterpiece of the Italian Renaissance, it has been described as "the best known ...
example pictured here provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a
shear mapping
In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of mappi ...
. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points ''along'' the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either.
Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
like
, in which case the eigenvectors are functions called
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s that are scaled by that differential operator, such as
Alternatively, the linear transformation could take the form of an ''n'' by ''n'' matrix, in which case the eigenvectors are ''n'' by 1 matrices. If the linear transformation is expressed in the form of an ''n'' by ''n'' matrix ''A'', then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication
where the eigenvector ''v'' is an ''n'' by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to
decompose the matrix—for example by
diagonalizing
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...
it.
Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix ''eigen-'' is applied liberally when naming them:
* The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation.
* The set of all eigenvectors of ''T'' corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace, or the characteristic space of ''T'' associated with that eigenvalue.
* If a set of eigenvectors of ''T'' forms a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of the domain of ''T'', then this basis is called an eigenbasis.
History
Eigenvalues are often introduced in the context of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
or
matrix theory
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begi ...
. Historically, however, they arose in the study of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s and
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s.
In the 18th century,
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
studied the rotational motion of a
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
, and discovered the importance of the
principal axes.
Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...](_blank)
saw how their work could be used to classify the
quadric surface
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
s, and generalized it to arbitrary dimensions. Cauchy also coined the term ''racine caractéristique'' (characteristic root), for what is now called ''eigenvalue''; his term survives in ''
characteristic equation''.
Later,
Joseph Fourier
Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...
used the work of Lagrange and
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
to solve the
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
by
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
in his famous 1822 book ''
Théorie analytique de la chaleur
Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harm ...
''.
Charles-François Sturm developed Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real
symmetric matrices
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with re ...
have real eigenvalues. This was extended by
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
in 1855 to what are now called
Hermitian matrices
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
.
Around the same time,
Francesco Brioschi
Francesco Brioschi (22 December 1824 – 13 December 1897) was an Italian mathematician.
Biography
Brioschi was born in Milan in 1824. He graduated from the Collegio Borromeo in 1847.
From 1850 he taught analytical mechanics in the University o ...
proved that the eigenvalues of
orthogonal matrices
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
lie on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
, and
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...
found the corresponding result for
skew-symmetric matrices
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...
. Finally,
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
clarified an important aspect in the
stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial diffe ...
started by Laplace, by realizing that
defective matrices can cause instability.
In the meantime,
Joseph Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called ''
Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form:
for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...
''.
Schwarz studied the first eigenvalue of
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
on general domains towards the end of the 19th century, while
Poincaré studied
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
a few years later.
At the start of the 20th century,
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
studied the eigenvalues of
integral operator
An integral operator is an operator that involves integration. Special instances are:
* The operator of integration itself, denoted by the integral symbol
* Integral linear operators, which are linear operators induced by bilinear forms invol ...
s by viewing the operators as infinite matrices. He was the first to use the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
word ''eigen'', which means "own",
to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when
Richard von Mises
Richard Edler von Mises (; 19 April 1883 – 14 July 1953) was an Austrian scientist and mathematician who worked on solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics and probability theory. He held the position of Gordon ...
published the
power method
In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A, the algorithm will produce a number \lambda, which is the greatest (in absolute value) eigenvalue of A, and a nonzero vect ...
. One of the most popular methods today, the
QR algorithm
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm was developed in the late 1950s by John G. F. Francis and by ...
, was proposed independently by
John G. F. Francis and
Vera Kublanovskaya in 1961.
Eigenvalues and eigenvectors of matrices
Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.
[Cornell University Department of Mathematics (2016]
''Lower-Level Courses for Freshmen and Sophomores''
Accessed on 2016-03-27.[University of Michigan Mathematics (2016]
''Math Course Catalogue''
. Accessed on 2016-03-27.
Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, which is especially common in numerical and computational applications.
![Eigenvalue equation](https://upload.wikimedia.org/wikipedia/commons/5/58/Eigenvalue_equation.svg)
Consider -dimensional vectors that are formed as a list of scalars, such as the three-dimensional vectors
These vectors are said to be
scalar multiples of each other, or
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of IBM ...
or
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
, if there is a scalar such that
In this case
.
Now consider the linear transformation of -dimensional vectors defined by an by matrix ,
or
where, for each row,
If it occurs that and are scalar multiples, that is if
then is an eigenvector of the linear transformation and the scale factor is the eigenvalue corresponding to that eigenvector. Equation () is the eigenvalue equation for the matrix .
Equation () can be stated equivalently as
where is the by
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
and 0 is the zero vector.
Eigenvalues and the characteristic polynomial
Equation () has a nonzero solution ''v''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the matrix is zero. Therefore, the eigenvalues of ''A'' are values of ''λ'' that satisfy the equation
Using the
Leibniz formula for determinants
In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A is an n \times n matrix, where a_ is the entry in the i-th row and j-th column ...
, the left-hand side of Equation () is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
function of the variable ''λ'' and the
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
of this polynomial is ''n'', the order of the matrix ''A''. Its
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s depend on the entries of ''A'', except that its term of degree ''n'' is always (−1)
''n''''λ''
''n''. This polynomial is called the ''
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
'' of ''A''. Equation () is called the ''characteristic equation'' or the ''secular equation'' of ''A''.
The
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomial ...
implies that the characteristic polynomial of an ''n''-by-''n'' matrix ''A'', being a polynomial of degree ''n'', can be
factored
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind ...
into the product of ''n'' linear terms,
where each ''λ''
''i'' may be real but in general is a complex number. The numbers ''λ''
1, ''λ''
2, ..., ''λ''
''n'', which may not all have distinct values, are roots of the polynomial and are the eigenvalues of ''A''.
As a brief example, which is described in more detail in the examples section later, consider the matrix
Taking the determinant of , the characteristic polynomial of ''A'' is
Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of ''A''. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of v in the equation In this example, the eigenvectors are any nonzero scalar multiples of
If the entries of the matrix ''A'' are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero imaginary parts. Similarly, the eigenvalues may be
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
s even if all the entries of ''A'' are
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s or even if they are all integers. However, if the entries of ''A'' are all
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s, which include the rationals, the eigenvalues are complex algebraic numbers.
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
s, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two import ...
at least one of the roots is real. Therefore, any
real matrix
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\beg ...
with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.
Algebraic multiplicity
Let ''λ''
''i'' be an eigenvalue of an ''n'' by ''n'' matrix ''A''. The algebraic multiplicity ''μ''
''A''(''λ''
''i'') of the eigenvalue is its
multiplicity as a root of the characteristic polynomial, that is, the largest integer ''k'' such that (''λ'' − ''λ''
''i'')
''k'' divides evenly that polynomial.
Suppose a matrix ''A'' has dimension ''n'' and ''d'' ≤ ''n'' distinct eigenvalues. Whereas Equation () factors the characteristic polynomial of ''A'' into the product of ''n'' linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product of ''d'' terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,
If ''d'' = ''n'' then the right-hand side is the product of ''n'' linear terms and this is the same as Equation (). The size of each eigenvalue's algebraic multiplicity is related to the dimension ''n'' as
If ''μ''
''A''(''λ''
''i'') = 1, then ''λ''
''i'' is said to be a ''simple eigenvalue''. If ''μ''
''A''(''λ''
''i'') equals the geometric multiplicity of ''λ''
''i'', ''γ''
''A''(''λ''
''i''), defined in the next section, then ''λ''
''i'' is said to be a ''semisimple eigenvalue''.
Eigenspaces, geometric multiplicity, and the eigenbasis for matrices
Given a particular eigenvalue ''λ'' of the ''n'' by ''n'' matrix ''A'', define the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''E'' to be all vectors v that satisfy Equation (),
On one hand, this set is precisely the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
or nullspace of the matrix (''A'' − ''λI''). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of ''A'' associated with ''λ''. So, the set ''E'' is the
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of the zero vector with the set of all eigenvectors of ''A'' associated with ''λ'', and ''E'' equals the nullspace of (''A'' − ''λI''). ''E'' is called the eigenspace or characteristic space of ''A'' associated with ''λ''. In general ''λ'' is a complex number and the eigenvectors are complex ''n'' by 1 matrices. A property of the nullspace is that it is a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
, so ''E'' is a linear subspace of
.
Because the eigenspace ''E'' is a linear subspace, it is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
under addition. That is, if two vectors u and v belong to the set ''E'', written , then or equivalently . This can be checked using the
distributive property
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
of matrix multiplication. Similarly, because ''E'' is a linear subspace, it is closed under scalar multiplication. That is, if and ''α'' is a complex number, or equivalently . This can be checked by noting that multiplication of complex matrices by complex numbers is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
. As long as u + v and ''α''v are not zero, they are also eigenvectors of ''A'' associated with ''λ''.
The dimension of the eigenspace ''E'' associated with ''λ'', or equivalently the maximum number of linearly independent eigenvectors associated with ''λ'', is referred to as the eigenvalue's geometric multiplicity ''γ''
''A''(''λ''). Because ''E'' is also the nullspace of (''A'' − ''λI''), the geometric multiplicity of ''λ'' is the dimension of the nullspace of (''A'' − ''λI''), also called the ''nullity'' of (''A'' − ''λI''), which relates to the dimension and rank of (''A'' − ''λI'') as
Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed ''n''.
To prove the inequality
, consider how the definition of geometric multiplicity implies the existence of
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
eigenvectors
, such that
. We can therefore find a (unitary) matrix
whose first
columns are these eigenvectors, and whose remaining columns can be any orthonormal set of
vectors orthogonal to these eigenvectors of
. Then
has full rank and is therefore invertible, and
with
a matrix whose top left block is the diagonal matrix
. This implies that
. In other words,
is similar to
, which implies that
. But from the definition of
we know that
contains a factor
, which means that the algebraic multiplicity of
must satisfy
.
Suppose
has
distinct eigenvalues
, where the geometric multiplicity of
is
. The total geometric multiplicity of
,
is the dimension of the
sum of all the eigenspaces of
's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of
. If
, then
* The direct sum of the eigenspaces of all of
's eigenvalues is the entire vector space
.
* A basis of
can be formed from
linearly independent eigenvectors of
; such a basis is called an eigenbasis
* Any vector in
can be written as a linear combination of eigenvectors of
.
Additional properties of eigenvalues
Let
be an arbitrary
matrix of complex numbers with eigenvalues
. Each eigenvalue appears
times in this list, where
is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:
* The
trace
Trace may refer to:
Arts and entertainment Music
* Trace (Son Volt album), ''Trace'' (Son Volt album), 1995
* Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* The Trace (album), ''The ...
of
, defined as the sum of its diagonal elements, is also the sum of all eigenvalues,
*:
* The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of
is the product of all its eigenvalues,
*:
* The eigenvalues of the
th power of
; i.e., the eigenvalues of
, for any positive integer
, are
.
* The matrix
is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
if and only if every eigenvalue is nonzero.
* If
is invertible, then the eigenvalues of
are
and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the
reciprocal polynomial
In algebra, given a polynomial
:p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n,
with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial,* denoted by or , is the polynomial
:p^*(x) = a_n + a_x + \cdots + a_0x^n = ...
of the original, the eigenvalues share the same algebraic multiplicity.
* If
is equal to its
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
, or equivalently if
is
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
, then every eigenvalue is real. The same is true of any
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
real matrix.
* If
is not only Hermitian but also
positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.
* If
is
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigroup ...
, every eigenvalue has absolute value
.
* if
is a
matrix and
are its eigenvalues, then the eigenvalues of matrix
(where
is the identity matrix) are
. Moreover, if
, the eigenvalues of
are
. More generally, for a polynomial
the eigenvalues of matrix
are
.
Left and right eigenvectors
Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a right eigenvector, namely a ''column'' vector that ''right'' multiplies the
matrix
in the defining equation, Equation (),
The eigenvalue and eigenvector problem can also be defined for ''row'' vectors that ''left'' multiply matrix
. In this formulation, the defining equation is
where
is a scalar and
is a
matrix. Any row vector
satisfying this equation is called a left eigenvector of
and
is its associated eigenvalue. Taking the transpose of this equation,
Comparing this equation to Equation (), it follows immediately that a left eigenvector of
is the same as the transpose of a right eigenvector of
, with the same eigenvalue. Furthermore, since the characteristic polynomial of
is the same as the characteristic polynomial of
, the eigenvalues of the left eigenvectors of
are the same as the eigenvalues of the right eigenvectors of
.
Diagonalization and the eigendecomposition
Suppose the eigenvectors of ''A'' form a basis, or equivalently ''A'' has ''n'' linearly independent eigenvectors v
1, v
2, ..., v
''n'' with associated eigenvalues ''λ''
1, ''λ''
2, ..., ''λ''
''n''. The eigenvalues need not be distinct. Define a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
''Q'' whose columns are the ''n'' linearly independent eigenvectors of ''A'',
:
Since each column of ''Q'' is an eigenvector of ''A'', right multiplying ''A'' by ''Q'' scales each column of ''Q'' by its associated eigenvalue,
:
With this in mind, define a diagonal matrix Λ where each diagonal element Λ
''ii'' is the eigenvalue associated with the ''i''th column of ''Q''. Then
:
Because the columns of ''Q'' are linearly independent, Q is invertible. Right multiplying both sides of the equation by ''Q''
−1,
:
or by instead left multiplying both sides by ''Q''
−1,
:
''A'' can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the
eigendecomposition
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matri ...
and it is a
similarity transformation. Such a matrix ''A'' is said to be ''similar'' to the diagonal matrix Λ or ''
diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
''. The matrix ''Q'' is the change of basis matrix of the similarity transformation. Essentially, the matrices ''A'' and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ.
Conversely, suppose a matrix ''A'' is diagonalizable. Let ''P'' be a non-singular square matrix such that ''P''
−1''AP'' is some diagonal matrix ''D''. Left multiplying both by ''P'', . Each column of ''P'' must therefore be an eigenvector of ''A'' whose eigenvalue is the corresponding diagonal element of ''D''. Since the columns of ''P'' must be linearly independent for ''P'' to be invertible, there exist ''n'' linearly independent eigenvectors of ''A''. It then follows that the eigenvectors of ''A'' form a basis if and only if ''A'' is diagonalizable.
A matrix that is not diagonalizable is said to be
defective
Defective may refer to::
*Defective matrix, in algebra
*Defective verb, in linguistics
*Defective, or ''haser'', in Hebrew orthography, a spelling variant that does not include mater lectionis
*Something presenting an anomaly, such as a product de ...
. For defective matrices, the notion of eigenvectors generalizes to
generalized eigenvector
In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.
Let V be an n-dimensional vector space; let \phi be a linear map ...
s and the diagonal matrix of eigenvalues generalizes to the
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to som ...
. Over an algebraically closed field, any matrix ''A'' has a
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to som ...
and therefore admits a basis of generalized eigenvectors and a decomposition into
generalized eigenspace
In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.
Let V be an n-dimensional vector space; let \phi be a linear map ...
s.
Variational characterization
In the
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
case, eigenvalues can be given a variational characterization. The largest eigenvalue of
is the maximum value of the
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
. A value of
that realizes that maximum, is an eigenvector.
Matrix examples
Two-dimensional matrix example
![Eigenvectors](https://upload.wikimedia.org/wikipedia/commons/0/06/Eigenvectors.gif)
Consider the matrix
The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors ''v'' of this transformation satisfy Equation (), and the values of ''λ'' for which the determinant of the matrix (''A'' − ''λI'') equals zero are the eigenvalues.
Taking the determinant to find characteristic polynomial of ''A'',
Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of ''A''.
For , Equation () becomes,
Any nonzero vector with ''v''
1 = −''v''
2 solves this equation. Therefore,
is an eigenvector of ''A'' corresponding to ''λ'' = 1, as is any scalar multiple of this vector.
For , Equation () becomes
Any nonzero vector with ''v''
1 = ''v''
2 solves this equation. Therefore,
is an eigenvector of ''A'' corresponding to ''λ'' = 3, as is any scalar multiple of this vector.
Thus, the vectors v
''λ''=1 and v
''λ''=3 are eigenvectors of ''A'' associated with the eigenvalues and , respectively.
Three-dimensional matrix example
Consider the matrix
The characteristic polynomial of ''A'' is
The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of ''A''. These eigenvalues correspond to the eigenvectors and or any nonzero multiple thereof.
Three-dimensional matrix example with complex eigenvalues
Consider the
cyclic permutation matrix
This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − ''λ''
3, whose roots are
where
is an
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
with
For the real eigenvalue ''λ''
1 = 1, any vector with three equal nonzero entries is an eigenvector. For example,
For the complex conjugate pair of imaginary eigenvalues,
Then
and
Therefore, the other two eigenvectors of ''A'' are complex and are
and
with eigenvalues ''λ''
2 and ''λ''
3, respectively. The two complex eigenvectors also appear in a complex conjugate pair,
Diagonal matrix example
Matrices with entries only along the main diagonal are called ''
diagonal matrices
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
''. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix
The characteristic polynomial of ''A'' is
which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of ''A''.
Each diagonal element corresponds to an eigenvector whose only nonzero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Triangular matrix example
A matrix whose elements above the main diagonal are all zero is called a ''lower
triangular matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
'', while a matrix whose elements below the main diagonal are all zero is called an ''upper triangular matrix''. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.
Consider the lower triangular matrix,
The characteristic polynomial of ''A'' is
which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of ''A''.
These eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Matrix with repeated eigenvalues example
As in the previous example, the lower triangular matrix
has a characteristic polynomial that is the product of its diagonal elements,
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The ''algebraic multiplicity'' of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is ''μ''
''A'' = 4 = ''n'', the order of the characteristic polynomial and the dimension of ''A''.
On the other hand, the ''geometric multiplicity'' of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector
and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector
. The total geometric multiplicity ''γ''
''A'' is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.
Eigenvector-eigenvalue identity
For a
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
, the norm squared of the ''j''th component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding
minor matrix,
where
is the
submatrix
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begin ...
formed by removing the ''j''th row and column from the original matrix. This identity also extends to
diagonalizable matrices
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...
, and has been rediscovered many times in the literature.
Eigenvalues and eigenfunctions of differential operators
The definitions of eigenvalue and eigenvectors of a linear transformation ''T'' remains valid even if the underlying vector space is an infinite-dimensional
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
or
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. A widely used class of linear transformations acting on infinite-dimensional spaces are the
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s on
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s. Let ''D'' be a linear differential operator on the space C
∞ of infinitely
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
real functions of a real argument ''t''. The eigenvalue equation for ''D'' is the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
The functions that satisfy this equation are eigenvectors of ''D'' and are commonly called eigenfunctions.
Derivative operator example
Consider the derivative operator
with eigenvalue equation
This differential equation can be solved by multiplying both sides by ''dt''/''f''(''t'') and integrating. Its solution, the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for ''λ'' = 0 the eigenfunction ''f''(''t'') is a constant.
The main
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
article gives other examples.
General definition
The concept of eigenvalues and eigenvectors extends naturally to arbitrary
linear transformations
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
on arbitrary vector spaces. Let ''V'' be any vector space over some
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K'' of
scalars
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
, and let ''T'' be a linear transformation mapping ''V'' into ''V'',
We say that a nonzero vector v ∈ ''V'' is an eigenvector of ''T'' if and only if there exists a scalar ''λ'' ∈ ''K'' such that
This equation is called the eigenvalue equation for ''T'', and the scalar ''λ'' is the eigenvalue of ''T'' corresponding to the eigenvector v. ''T''(v) is the result of applying the transformation ''T'' to the vector v, while ''λ''v is the product of the scalar ''λ'' with v.
Eigenspaces, geometric multiplicity, and the eigenbasis
Given an eigenvalue ''λ'', consider the set
which is the union of the zero vector with the set of all eigenvectors associated with ''λ''. ''E'' is called the eigenspace or characteristic space of ''T'' associated with ''λ''.
By definition of a linear transformation,
for x, y ∈ ''V'' and ''α'' ∈ ''K''. Therefore, if u and v are eigenvectors of ''T'' associated with eigenvalue ''λ'', namely u, v ∈ ''E'', then
So, both u + v and αv are either zero or eigenvectors of ''T'' associated with ''λ'', namely u + v, ''α''v ∈ ''E'', and ''E'' is closed under addition and scalar multiplication. The eigenspace ''E'' associated with ''λ'' is therefore a linear subspace of ''V''.
If that subspace has dimension 1, it is sometimes called an eigenline.
The geometric multiplicity ''γ''
''T''(''λ'') of an eigenvalue ''λ'' is the dimension of the eigenspace associated with ''λ'', i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue. By the definition of eigenvalues and eigenvectors, ''γ''
''T''(''λ'') ≥ 1 because every eigenvalue has at least one eigenvector.
The eigenspaces of ''T'' always form a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
. As a consequence, eigenvectors of ''different'' eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension ''n'' of the vector space on which ''T'' operates, and there cannot be more than ''n'' distinct eigenvalues.
Any subspace spanned by eigenvectors of ''T'' is an
invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''.
General descrip ...
of ''T'', and the restriction of ''T'' to such a subspace is diagonalizable. Moreover, if the entire vector space ''V'' can be spanned by the eigenvectors of ''T'', or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of ''T'' is the entire vector space ''V'', then a basis of ''V'' called an eigenbasis can be formed from linearly independent eigenvectors of ''T''. When ''T'' admits an eigenbasis, ''T'' is diagonalizable.
Spectral theory
If ''λ'' is an eigenvalue of ''T'', then the operator (''T'' − ''λI'') is not one-to-one, and therefore its inverse (''T'' − ''λI'')
−1 does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (''T'' − ''λI'') may not have an inverse even if ''λ'' is not an eigenvalue.
For this reason, in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
eigenvalues can be generalized to the
spectrum of a linear operator ''T'' as the set of all scalars ''λ'' for which the operator (''T'' − ''λI'') has no
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.
Associative algebras and representation theory
One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an
algebra representation
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functo ...
– an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
acting on a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
. The study of such actions is the field of
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
.
The
representation-theoretical concept of weight is an analog of eigenvalues, while ''weight vectors'' and ''weight spaces'' are the analogs of eigenvectors and eigenspaces, respectively.
Dynamic equations
The simplest
difference equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s have the form
:
The solution of this equation for ''x'' in terms of ''t'' is found by using its characteristic equation
:
which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the ''k'' – 1 equations
giving a ''k''-dimensional system of the first order in the stacked variable vector
in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives ''k'' characteristic roots
for use in the solution equation
:
A similar procedure is used for solving a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
of the form
:
Calculation
The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.
Classical method
The classical method is to first find the eigenvalues, and then calculate the eigenvectors for each eigenvalue. It is in several ways poorly suited for non-exact arithmetics such as
floating-point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...
.
Eigenvalues
The eigenvalues of a matrix
can be determined by finding the roots of the characteristic polynomial. This is easy for
matrices, but the difficulty increases rapidly with the size of the matrix.
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required
accuracy
Accuracy and precision are two measures of ''observational error''.
''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each other ...
. However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable
round-off error
A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
s, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by
Wilkinson's polynomial
In numerical analysis, Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial: the location of the roots can be very sensitive to perturbatio ...
). Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
, which for an
matrix is a sum of
different products.
Explicit
algebraic formulas for the roots of a polynomial exist only if the degree
is 4 or less. According to the
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means th ...
there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree
is the characteristic polynomial of some
companion matrix In linear algebra, the Frobenius companion matrix of the monic polynomial
:
p(t)=c_0 + c_1 t + \cdots + c_t^ + t^n ~,
is the square matrix defined as
:C(p)=\begin
0 & 0 & \dots & 0 & -c_0 \\
1 & 0 & \dots & 0 & -c_1 \\
0 & 1 & \dots & 0 & -c_2 ...
of order
.) Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate
numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Mathem ...
s. Even the
exact formula for the roots of a degree 3 polynomial is numerically impractical.
Eigenvectors
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that becomes a
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three ...
with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
we can find its eigenvectors by solving the equation
, that is
This matrix equation is equivalent to two
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s
that is
Both equations reduce to the single linear equation
. Therefore, any vector of the form
, for any nonzero real number
, is an eigenvector of
with eigenvalue
.
The matrix
above has another eigenvalue
. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of
, that is, any vector of the form
, for any nonzero real number
.
Simple iterative methods
The converse approach, of first seeking the eigenvectors and then determining each eigenvalue from its eigenvector, turns out to be far more tractable for computers. The easiest algorithm here consists of picking an arbitrary starting vector and then repeatedly multiplying it with the matrix (optionally normalizing the vector to keep its elements of reasonable size); this makes the vector converge towards an eigenvector.
A variation is to instead multiply the vector by this causes it to converge to an eigenvector of the eigenvalue closest to
If
is (a good approximation of) an eigenvector of
, then the corresponding eigenvalue can be computed as
:
where
denotes the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
of
.
Modern methods
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the
QR algorithm
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm was developed in the late 1950s by John G. F. Francis and by ...
was designed in 1961. Combining the
Householder transformation
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformatio ...
with the LU decomposition results in an algorithm with better convergence than the QR algorithm. For large
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
sparse matrices
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse b ...
, the
Lanczos algorithm
The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power iteration, power methods to find the m "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an n \times n ...
is one example of an efficient
iterative method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived fr ...
to compute eigenvalues and eigenvectors, among several other possibilities.
Most numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation, although sometimes implementors choose to discard the eigenvector information as soon as it is no longer needed.
Applications
Eigenvalues of geometric transformations
The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.
{, class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
, + Eigenvalues of geometric transformations
, -
!
! scope="col" ,
Scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
! scope="col" , Unequal scaling
! scope="col" ,
Rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
! scope="col" ,
Horizontal shear
! scope="col" ,
Hyperbolic rotation
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping.
For a fixed positive real number , th ...
, -
! scope="row" , Illustration
,
![Homothety in two dim](https://upload.wikimedia.org/wikipedia/commons/2/2b/Homothety_in_two_dim.svg)
,
![Unequal scaling](https://upload.wikimedia.org/wikipedia/commons/b/be/Unequal_scaling.svg)
,
![Rotation](https://upload.wikimedia.org/wikipedia/commons/c/c9/Rotation.png)
,
![Shear](https://upload.wikimedia.org/wikipedia/commons/2/2a/Shear.svg)
,
![Squeeze r=1](https://upload.wikimedia.org/wikipedia/commons/6/67/Squeeze_r%3D1.5.svg)
, - style="vertical-align:top"
! scope="row" , Matrix
,
,
,
,
,
, -
! scope="row" , Characteristic
polynomial
,
,
,
,
,
, -
! scope="row" , Eigenvalues,
,
,
,
,
,
, -
! scope="row" , Algebraic ,
,
,
,
,
,
, -
! scope="row" , Geometric ,
,
,
,
,
,
, -
! scope="row" , Eigenvectors
, All nonzero vectors
,
,
,
,
The characteristic equation for a rotation is a
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
with
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
, which is a negative number whenever is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers,
; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.
A linear transformation that takes a square to a rectangle of the same area (a
squeeze mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping.
For a fixed positive real number , th ...
) has reciprocal eigenvalues.
Schrödinger equation
![HAtomOrbitals](https://upload.wikimedia.org/wikipedia/commons/c/cf/HAtomOrbitals.png)
An example of an eigenvalue equation where the transformation
is represented in terms of a differential operator is the time-independent
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
:
:
where
, the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
, is a second-order
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
and
, the
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
, is one of its eigenfunctions corresponding to the eigenvalue
, interpreted as its
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
.
However, in the case where one is interested only in the
bound state
Bound or bounds may refer to:
Mathematics
* Bound variable
* Upper and lower bounds, observed limits of mathematical functions
Physics
* Bound state, a particle that has a tendency to remain localized in one or more regions of space
Geography
*B ...
solutions of the Schrödinger equation, one looks for
within the space of
square integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
functions. Since this space is a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
with a well-defined
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
, one can introduce a
basis set in which
and
can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.
The
bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by
. In this notation, the Schrödinger equation is:
:
where
is an eigenstate of
and
represents the eigenvalue.
is an
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above
is understood to be the vector obtained by application of the transformation
to
.
Wave transport
Light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
,
acoustic wave
Acoustic waves are a type of energy propagation through a medium by means of adiabatic loading and unloading. Important quantities for describing acoustic waves are acoustic pressure, particle velocity, particle displacement and acoustic intensit ...
s, and
microwave
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
s are randomly
scattered
Scattered may refer to:
Music
* ''Scattered'' (album), a 2010 album by The Handsome Family
* "Scattered" (The Kinks song), 1993
* "Scattered", a song by Ace Young
* "Scattered", a song by Lauren Jauregui
* "Scattered", a song by Green Day from ' ...
numerous times when traversing a static
disordered system. Even though multiple scattering repeatedly randomizes the waves, ultimately coherent wave transport through the system is a deterministic process which can be described by a field transmission matrix
. The eigenvectors of the transmission operator
form a set of disorder-specific input wavefronts which enable waves to couple into the disordered system's eigenchannels: the independent pathways waves can travel through the system. The eigenvalues,
, of
correspond to the intensity transmittance associated with each eigenchannel. One of the remarkable properties of the transmission operator of diffusive systems is their bimodal eigenvalue distribution with
and
. Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.
Molecular orbitals
In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, and in particular in
atomic and
molecular physics
Molecular physics is the study of the physical properties of molecules and molecular dynamics. The field overlaps significantly with physical chemistry, chemical physics, and quantum chemistry. It is often considered as a sub-field of atomic, mo ...
, within the
Hartree–Fock theory, the
atomic and
molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding ...
s can be defined by the eigenvectors of the
Fock operator. The corresponding eigenvalues are interpreted as
ionization potential
Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule i ...
s via
Koopmans' theorem
Koopmans' theorem states that in closed-shell Hartree–Fock theory (HF), the first ionization energy of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO). This theorem is named afte ...
. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Such equations are usually solved by an
iteration
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
procedure, called in this case
self-consistent field
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
method. In
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
, one often represents the Hartree–Fock equation in a non-
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
basis set. This particular representation is a
generalized eigenvalue problem
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matri ...
called
Roothaan equations The Roothaan equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussian-type or Slater-type. It applies to closed-shell molecules or atoms where all molecular orbitals or atomic orbitals ...
.
Geology and glaciology
In
geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
, especially in the study of
glacial till
image:Geschiebemergel.JPG, Closeup of glacial till. Note that the larger grains (pebbles and gravel) in the till are completely surrounded by the matrix of finer material (silt and sand), and this characteristic, known as ''matrix support'', is d ...
, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of
clasts
Clastic rocks are composed of fragments, or clasts, of pre-existing minerals and rock. A clast is a fragment of geological detritus,Essentials of Geology, 3rd Ed, Stephen Marshak, p. G-3 chunks, and smaller grains of rock broken off other rocks ...
in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram, or as a Stereonet on a Wulff Net.
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered
by their eigenvalues
;
then is the primary orientation/dip of clast,
is the secondary and
is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a
compass rose
A compass rose, sometimes called a wind rose, rose of the winds or compass star, is a figure on a compass, map, nautical chart, or monument used to display the orientation of the cardinal directions (north, east, south, and west) and their int ...
of
360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of
,
, and
are dictated by the nature of the sediment's fabric. If
, the fabric is said to be isotropic. If
, the fabric is said to be planar. If
, the fabric is said to be linear.
Principal component analysis
![GaussianScatterPCA](https://upload.wikimedia.org/wikipedia/commons/1/15/GaussianScatterPCA.png)
The
eigendecomposition
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matri ...
of a
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
positive semidefinite (PSD)
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
yields an
orthogonal basis In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis ...
of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in
multivariate analysis
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the dif ...
, where the
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of s ...
covariance matrices
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
are PSD. This orthogonal decomposition is called
principal component analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
(PCA) in statistics. PCA studies
linear relation
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.
More precisely, if e_1,\dots,e_n are elements of a (left) module over a ring ( ...
s among variables. PCA is performed on the
covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
or the
correlation matrix
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...
(in which each variable is scaled to have its
sample variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
equal to one). For the covariance or correlation matrix, the eigenvectors correspond to
principal components
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
and the eigenvalues to the
variance explained by the principal components. Principal component analysis of the correlation matrix provides an
orthogonal basis In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis ...
for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data.
Principal component analysis is used as a means of
dimensionality reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally ...
in the study of large
data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...
s, such as those encountered in
bioinformatics
Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...
. In
Q methodology Q methodology is a research method used in psychology and in social sciences to study people's "subjectivity"—that is, their viewpoint. Q was developed by psychologist William Stephenson. It has been used both in clinical settings for assessing a ...
, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of ''practical'' significance (which differs from the
statistical significance
In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...
of
hypothesis testing
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
; cf.
criteria for determining the number of factors). More generally, principal component analysis can be used as a method of
factor analysis
Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed ...
in
structural equation model
Structural equation modeling (SEM) is a label for a diverse set of methods used by scientists in both experimental and observational research across the sciences, business, and other fields. It is used most in the social and behavioral scienc ...
ing.
Vibration analysis
![Mode Shape of a Tuning Fork at Eigenfrequency 440](https://upload.wikimedia.org/wikipedia/commons/8/82/Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif)
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by
or
that is, acceleration is proportional to position (i.e., we expect
to be sinusoidal in time).
In
dimensions,
becomes a
mass matrix
In analytical mechanics, the mass matrix is a symmetric matrix that expresses the connection between the time derivative \mathbf\dot q of the generalized coordinate vector of a system and the kinetic energy of that system, by the equation
:T ...
and
a
stiffness matrix
In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution ...
. Admissible solutions are then a linear combination of solutions to the
generalized eigenvalue problem
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matri ...
where
is the eigenvalue and
is the (imaginary)
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
. The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of
alone. Furthermore,
damped vibration, governed by
leads to a so-called
quadratic eigenvalue problem In mathematics, the quadratic eigenvalue problemF. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM
Rev., 43 (2001), pp. 235–286. (QEP), is to find scalar eigenvalues \lambda, left eigenvectors y and right eigenvectors x such that ...
,
This can be reduced to a generalized eigenvalue problem by
algebraic manipulation at the cost of solving a larger system.
The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using
finite element analysis
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
, but neatly generalize the solution to scalar-valued vibration problems.
Eigenfaces
![Eigenfaces](https://upload.wikimedia.org/wikipedia/commons/6/67/Eigenfaces.png)
In
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
, processed images of faces can be seen as vectors whose components are the
brightness
Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target. The perception is not linear to luminance, ...
es of each
pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device.
In most digital display devices, pixels are the smal ...
. The dimension of this vector space is the number of pixels. The eigenvectors of the
covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
associated with a large set of normalized pictures of faces are called
eigenface
An eigenface () is the name given to a set of eigenvectors when used in the computer vision problem of human face recognition. The approach of using eigenfaces for recognition was developed by Sirovich and Kirby and used by Matthew Turk and Alex ...
s; this is an example of
principal component analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
. They are very useful for expressing any face image as a
linear combination of some of them. In the
facial recognition branch of
biometrics
Biometrics are body measurements and calculations related to human characteristics. Biometric authentication (or realistic authentication) is used in computer science as a form of identification and access control. It is also used to identify in ...
, eigenfaces provide a means of applying
data compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression ...
to faces for
identification
Identification or identify may refer to:
*Identity document, any document used to verify a person's identity
Arts, entertainment and media
* ''Identify'' (album) by Got7, 2014
* "Identify" (song), by Natalie Imbruglia, 1999
* Identification ( ...
purposes. Research related to eigen vision systems determining hand gestures has also been made.
Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems for speaker adaptation.
Tensor of moment of inertia
In
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, the eigenvectors of the
moment of inertia tensor
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accelera ...
define the
principal axes of a
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
. The
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
of moment of
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
is a key quantity required to determine the rotation of a rigid body around its
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
.
Stress tensor
In
solid mechanics
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
, the
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
tensor is symmetric and so can be decomposed into a
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no
shear
Shear may refer to:
Textile production
*Animal shearing, the collection of wool from various species
**Sheep shearing
*The removal of nap during wool cloth production
Science and technology Engineering
*Shear strength (soil), the shear strength ...
components; the components it does have are the principal components.
Graphs
In
spectral graph theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix ...
, an eigenvalue of a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
is defined as an eigenvalue of the graph's
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
, or (increasingly) of the graph's
Laplacian matrix
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplac ...
due to its
discrete Laplace operator
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a Graph (discrete mathematics), graph or a lattice (group), discrete grid. For the case of a finite-dimensional graph ...
, which is either
(sometimes called the ''combinatorial Laplacian'') or
(sometimes called the ''normalized Laplacian''), where
is a diagonal matrix with
equal to the degree of vertex
, and in
, the
th diagonal entry is
. The
th principal eigenvector of a graph is defined as either the eigenvector corresponding to the
th largest or
th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
The principal eigenvector is used to measure the
centrality
In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a social network, key ...
of its vertices. An example is
Google
Google LLC () is an American multinational technology company focusing on search engine technology, online advertising, cloud computing, computer software, quantum computing, e-commerce, artificial intelligence, and consumer electronics. ...
's
PageRank
PageRank (PR) is an algorithm used by Google Search to rank web pages in their search engine results. It is named after both the term "web page" and co-founder Larry Page. PageRank is a way of measuring the importance of website pages. According ...
algorithm. The principal eigenvector of a modified
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the
stationary distribution Stationary distribution may refer to:
* A special distribution for a Markov chain such that if the chain starts with its stationary distribution, the marginal distribution of all states at any time will always be the stationary distribution. Assum ...
of the
Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via
spectral clustering
In multivariate statistics, spectral clustering techniques make use of the Spectrum of a matrix, spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity ...
. Other methods are also available for clustering.
Basic reproduction number
The basic reproduction number (
) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then
is the average number of people that one typical infectious person will infect. The generation time of an infection is the time,
, from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time
has passed.
is then the largest eigenvalue of the next generation matrix.
See also
*
Antieigenvalue theory In applied mathematics, antieigenvalue theory was developed by Karl Gustafson from 1966 to 1968. The theory is applicable to numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbol ...
*
Eigenoperator
In mathematics, an eigenoperator, ''A'', of a matrix ''H'' is a linear operator such that
: ,A= \lambda A \,
where \lambda is a corresponding scalar called an eigenvalue
In linear algebra, an eigenvector () or characteristic vector of ...
*
Eigenplane
In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. By analogy with the term ''eigenvector'' for a vector which, when operated on by a linear operator is another vector which is a scalar multiple of itsel ...
*
Eigenmoments
EigenMoments is a set of orthogonal, noise robust, invariant to rotation, scaling and translation and distribution sensitive moments. Their application can be found in signal processing and computer vision as descriptors of the signal or image ...
*
Eigenvalue algorithm
In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.
Eigenvalues and eigenvectors
Given an squa ...
*
Introduction to eigenstates
Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. The uncertainty principle also says that eliminating un ...
*
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to som ...
*
List of numerical-analysis software
Listed here are notable end-user computer applications intended for use with numerical or data analysis:
Numerical-software packages
General-purpose computer algebra systems
Interface-oriented
Language-oriented
Historically significa ...
*
Nonlinear eigenproblem
In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form
...
*
Normal eigenvalue
In mathematics, specifically in spectral theory, an eigenvalue of a closed linear operator is called normal if the space admits a decomposition into a direct sum of a finite-dimensional generalized eigenspace and an invariant subspace where A-\lam ...
*
Quadratic eigenvalue problem In mathematics, the quadratic eigenvalue problemF. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM
Rev., 43 (2001), pp. 235–286. (QEP), is to find scalar eigenvalues \lambda, left eigenvectors y and right eigenvectors x such that ...
*
Singular value In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self- ...
*
Spectrum of a matrix In mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if T\colon V \to V is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars \lambda such that T-\lambda I is not invertib ...
Notes
Citations
Sources
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External links
What are Eigen Values?– non-technical introduction from PhysLink.com's "Ask the Experts"
– Tutorial and Interactive Program from Revoledu.
Introduction to Eigen Vectors and Eigen Values– lecture from Khan Academy
Eigenvectors and eigenvalues , Essence of linear algebra, chapter 10– A visual explanation with
3Blue1Brown
3Blue1Brown is a math YouTube channel created and run by Grant Sanderson. The channel focuses on teaching higher mathematics from a visual perspective, and on the process of discovery and inquiry-based learning in mathematics, which Sanderson cal ...
Matrix Eigenvectors Calculatorfrom Symbolab (Click on the bottom right button of the 2x12 grid to select a matrix size. Select an
size (for a square matrix), then fill out the entries numerically and click on the Go button. It can accept complex numbers as well.)
Theory
Edited by Zhaojun Bai,
James Demmel
James Weldon Demmel Jr. (born October 19, 1955) is an American mathematician and computer scientist, the ''Dr. Richard Carl Dehmel Distinguished Professor of Mathematics and Computer Science'' at the University of California, Berkeley.
In 1999, ...
, Jack Dongarra, Axel Ruhe, and
Henk van der Vorst
Hendrik "Henk" Albertus van der Vorst (born 5 May 1944, Venlo) is a Dutch mathematician and Emeritus Professor of Numerical Analysis at Utrecht University. According to the Institute for Scientific Information (ISI), his paper
on the BiCGSTAB ...
{{DEFAULTSORT:Eigenvalues And Eigenvectors
Abstract algebra
Linear algebra
Mathematical physics
Matrix theory
Singular value decomposition