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

\lambda

, 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

T(\mathbf) = \lambda \mathbf,

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

A\mathbf = \lambda \mathbf.

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 **Ge ...

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 fo ...

, 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.) ...

. 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

T(\mathbf) = \lambda \mathbf,

referred to as the eigenvalue equation or eigenequation. In general, ''λ'' may be any

. 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 ...

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 like

\tfrac

, 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

\frace^ = \lambda e^.

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

A\mathbf v = \lambda \mathbf v,

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

of the domain of ''T'', then this basis is called an eigenbasis.

History

Eigenvalues are often introduced in the context of

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 LagrangiaAugustin-Louis Cauchy saw how their work could be used to classify the quadric surfaces, 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 ...

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 ...

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 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 t ...

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

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 Gordo ...

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 vec ...

. 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

Consider -dimensional vectors that are formed as a list of scalars, such as the three-dimensional vectors

\mathbf x = \begin1\\-3\\4\end\quad\mbox\quad \mathbf y = \begin-20\\60\\-80\end.

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 ...

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

\mathbf x = \lambda \mathbf y.

In this case

\lambda = -\frac

. Now consider the linear transformation of -dimensional vectors defined by an by matrix ,

A \mathbf v = \mathbf w,

\begin
    A_ & A_ & \cdots & A_ \\
    A_ & A_ & \cdots & A_ \\
    \vdots & \vdots & \ddots & \vdots \\
    A_ & A_ & \cdots & A_ \\
  \end\begin
    v_1 \\ v_2 \\ \vdots \\ v_n
  \end = \begin
    w_1 \\ w_2 \\ \vdots \\ w_n
  \end

where, for each row,

w_i = A_ v_1 + A_ v_2 + \cdots + A_ v_n = \sum_^n A_ v_j.

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 polynomia ...

implies that the characteristic polynomial of an ''n''-by-''n'' matrix ''A'', being a polynomial of degree ''n'', can be factored 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

A = \begin
  2 & 1\\
  1 & 2
\end.

Taking the determinant of , the characteristic polynomial of ''A'' is

, A - \lambda I,  = \begin
    2 - \lambda & 1 \\
    1           & 2 - \lambda
  \end =
  3 - 4\lambda + \lambda^2.

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

\mathbf v_ = \begin 1 \\ -1 \end, \quad \mathbf v_ = \begin 1 \\ 1 \end.

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 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,

, A - \lambda I,  = (\lambda_1 - \lambda)^(\lambda_2 - \lambda)^ \cdots (\lambda_d - \lambda)^.

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

\begin
      1 &\leq \mu_A(\lambda_i) \leq n, \\
  \mu_A &= \sum_^d \mu_A\left(\lambda_i\right) = n.
\end

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 (),

E = \left\.

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

\mathbb^n

. Because the eigenspace ''E'' is a linear subspace, it is closed 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

\gamma_A(\lambda) = n - \operatorname(A - \lambda I).

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''.

1 \le \gamma_A(\lambda) \le \mu_A(\lambda) \le n

To prove the inequality

\gamma_A(\lambda)\le\mu_A(\lambda)

, consider how the definition of geometric multiplicity implies the existence of

\gamma_A(\lambda)

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

\boldsymbol_1,\, \ldots,\, \boldsymbol_

, such that

A \boldsymbol_k = \lambda \boldsymbol_k

. We can therefore find a (unitary) matrix

V

whose first

\gamma_A(\lambda)

columns are these eigenvectors, and whose remaining columns can be any orthonormal set of

n - \gamma_A(\lambda)

vectors orthogonal to these eigenvectors of

A

. Then

V

has full rank and is therefore invertible, and

AV=VD

with

D

a matrix whose top left block is the diagonal matrix

\lambda I_

. This implies that

(A - \xi I)V = V(D - \xi I)

. In other words,

A - \xi I

is similar to

D - \xi I

, which implies that

\det(A - \xi I) = \det(D - \xi I)

. But from the definition of

D

we know that

\det(D - \xi I)

contains a factor

(\xi - \lambda)^

, which means that the algebraic multiplicity of

\lambda

must satisfy

\mu_A(\lambda) \ge \gamma_A(\lambda)

. Suppose

A

has

d \leq n

distinct eigenvalues

\lambda_1, \ldots, \lambda_d

, where the geometric multiplicity of

\lambda_i

\gamma_A (\lambda_i)

. The total geometric multiplicity of

A

\begin
  \gamma_A &= \sum_^d \gamma_A(\lambda_i), \\
         d &\le \gamma_A \le n,
\end

is the dimension of the sum of all the eigenspaces of

A

's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of

A

. If

\gamma_A=n

, then * The direct sum of the eigenspaces of all of

A

's eigenvalues is the entire vector space

\mathbb^n

. * A basis of

\mathbb^n

can be formed from

n

linearly independent eigenvectors of

A

; such a basis is called an eigenbasis * Any vector in

\mathbb^n

can be written as a linear combination of eigenvectors of

A

Additional properties of eigenvalues

Let

A

be an arbitrary

n \times n

matrix of complex numbers with eigenvalues

\lambda_1, \ldots, \lambda_n

. Each eigenvalue appears

\mu_A(\lambda_i)

times in this list, where

\mu_A(\lambda_i)

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), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...

A

, defined as the sum of its diagonal elements, is also the sum of all eigenvalues, *:

\operatorname(A) = \sum_^n a_ = \sum_^n \lambda_i = \lambda_1 + \lambda_2 + \cdots + \lambda_n.

* The

A

is the product of all its eigenvalues, *:

\det(A) = \prod_^n \lambda_i = \lambda_1\lambda_2 \cdots \lambda_n.

* The eigenvalues of the

k

^th power of

A

; i.e., the eigenvalues of

A^k

, for any positive integer

k

, are

\lambda_1^k, \ldots, \lambda_n^k

. * The matrix

A

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

A

is invertible, then the eigenvalues of

A^

are

\frac, \ldots, \frac

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

A

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 ...

A^*

, or equivalently if

A

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 m ...

, 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

A

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

A

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 semigrou ...

, every eigenvalue has absolute value

, \lambda_i, =1

. * if

A

is a

n\times n

matrix and

\

are its eigenvalues, then the eigenvalues of matrix

I+A

(where

I

is the identity matrix) are

\

. Moreover, if

\alpha\in\mathbb C

, the eigenvalues of

\alpha I+A

are

\

. More generally, for a polynomial

P

the eigenvalues of matrix

P(A)

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

n \times n

matrix

A

in the defining equation, Equation (),

A \mathbf v = \lambda \mathbf v.

The eigenvalue and eigenvector problem can also be defined for ''row'' vectors that ''left'' multiply matrix

A

. In this formulation, the defining equation is

\mathbf u A = \kappa \mathbf u,

where

\kappa

is a scalar and

u

is a

1 \times n

matrix. Any row vector

u

satisfying this equation is called a left eigenvector of

A

and

\kappa

is its associated eigenvalue. Taking the transpose of this equation,

A^\textsf \mathbf u^\textsf = \kappa \mathbf u^\textsf.

Comparing this equation to Equation (), it follows immediately that a left eigenvector of

A

is the same as the transpose of a right eigenvector of

A^\textsf

, with the same eigenvalue. Furthermore, since the characteristic polynomial of

A^\textsf

is the same as the characteristic polynomial of

A

, the eigenvalues of the left eigenvectors of

A

are the same as the eigenvalues of the right eigenvectors of

A^\textsf

Diagonalization and the eigendecomposition

Suppose the eigenvectors of ''A'' form a basis, or equivalently ''A'' has ''n'' linearly independent eigenvectors v₁, v₂, ..., 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'', :

Q = \begin \mathbf v_1 & \mathbf v_2 & \cdots & \mathbf v_n \end.

Since each column of ''Q'' is an eigenvector of ''A'', right multiplying ''A'' by ''Q'' scales each column of ''Q'' by its associated eigenvalue, :

AQ = \begin \lambda_1 \mathbf v_1 & \lambda_2 \mathbf v_2 & \cdots & \lambda_n \mathbf v_n \end.

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 :

AQ = Q\Lambda.

Because the columns of ''Q'' are linearly independent, Q is invertible. Right multiplying both sides of the equation by ''Q''⁻¹, :

A = Q\Lambda Q^,

or by instead left multiplying both sides by ''Q''⁻¹, :

Q^AQ = \Lambda.

''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 matr ...

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 so ...

. Over an algebraically closed field, any matrix ''A'' has a

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 ...

Variational characterization

In the

case, eigenvalues can be given a variational characterization. The largest eigenvalue of

H

is the maximum value of the

\mathbf x^\textsf H \mathbf x / \mathbf x^\textsf \mathbf x

. A value of

\mathbf x

that realizes that maximum, is an eigenvector.

Matrix examples

Two-dimensional matrix example

Consider the matrix

A = \begin
  2 & 1\\
  1 & 2
\end.

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'',

&= (\lambda - 3)(\lambda - 1). \end

Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of ''A''. For , Equation () becomes,

(A - I)\mathbf_ = \begin 1 & 1\\ 1 & 1\end\beginv_1 \\ v_2\end = \begin0 \\ 0\end

1v_1 + 1v_2 = 0

Any nonzero vector with ''v''₁ = −''v''₂ solves this equation. Therefore,

\mathbf_ = \begin v_1 \\ -v_1 \end = \begin 1 \\ -1 \end

is an eigenvector of ''A'' corresponding to ''λ'' = 1, as is any scalar multiple of this vector. For , Equation () becomes

\begin
  (A - 3I)\mathbf_ &=
  \begin -1 & 1\\ 1 & -1 \end
    \begin v_1 \\ v_2 \end =
    \begin 0 \\ 0 \end \\
  -1v_1 + 1v_2 &= 0;\\
   1v_1 - 1v_2 &= 0
\end

Any nonzero vector with ''v''₁ = ''v''₂ solves this equation. Therefore,

\mathbf v_ = \begin v_1 \\ v_1 \end = \begin 1 \\ 1 \end

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

A = \begin
  2 & 0 & 0 \\
  0 & 3 & 4 \\
  0 & 4 & 9
\end.

The characteristic polynomial of ''A'' is

= -\lambda^3 + 14\lambda^2 - 35\lambda + 22. \end

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

A = \begin
  0 & 1 & 0\\
  0 & 0 & 1\\
  1 & 0 & 0
\end.

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

\begin
  \lambda_1 &= 1 \\
  \lambda_2 &= -\frac + i \frac \\
  \lambda_3 &= \lambda_2^* = -\frac - i \frac
\end

where

i

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,

A \begin 5\\ 5\\ 5 \end =
    \begin 5\\ 5\\ 5 \end =
    1 \cdot \begin 5\\ 5\\ 5 \end.

For the complex conjugate pair of imaginary eigenvalues,

\lambda_2\lambda_3 = 1, \quad \lambda_2^2 = \lambda_3, \quad \lambda_3^2 = \lambda_2.

Then

A \begin 1 \\ \lambda_2 \\ \lambda_3 \end =
    \begin \lambda_2 \\ \lambda_3 \\ 1 \end =
    \lambda_2 \cdot \begin 1 \\ \lambda_2 \\ \lambda_3 \end,

and

A \begin 1 \\ \lambda_3 \\ \lambda_2 \end =
    \begin \lambda_3 \\ \lambda_2 \\ 1 \end =
    \lambda_3 \cdot \begin 1 \\ \lambda_3 \\ \lambda_2 \end.

Therefore, the other two eigenvectors of ''A'' are complex and are

\mathbf v_ = \begin 1 & \lambda_2 & \lambda_3\end^\textsf

and

\mathbf v_ = \begin 1 & \lambda_3 & \lambda_2\end^\textsf

with eigenvalues ''λ''₂ and ''λ''₃, respectively. The two complex eigenvectors also appear in a complex conjugate pair,

\mathbf v_ = \mathbf v_^*.

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 m ...

''. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix

A = \begin 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3\end.

The characteristic polynomial of ''A'' is

, A - \lambda I,  = (1 - \lambda)(2 - \lambda)(3 - \lambda),

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,

\mathbf v_ = \begin 1\\ 0\\ 0 \end,\quad
  \mathbf v_ = \begin 0\\ 1\\ 0 \end,\quad
  \mathbf v_ = \begin 0\\ 0\\ 1 \end,

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,

A = \begin
  1 & 0 & 0\\
  1 & 2 & 0\\
  2 & 3 & 3
\end.

The characteristic polynomial of ''A'' is

, A - \lambda I,  = (1 - \lambda)(2 - \lambda)(3 - \lambda),

which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of ''A''. These eigenvalues correspond to the eigenvectors,

\mathbf v_ = \begin 1\\ -1\\  \frac\end,\quad
  \mathbf v_ = \begin 0\\  1\\ -3\end,\quad
  \mathbf v_ = \begin 0\\  0\\  1\end,

respectively, as well as scalar multiples of these vectors.

Matrix with repeated eigenvalues example

As in the previous example, the lower triangular matrix

A = \begin
  2 & 0 & 0 & 0 \\
  1 & 2 & 0 & 0 \\
  0 & 1 & 3 & 0  \\
  0 & 0 & 1 & 3
\end,

has a characteristic polynomial that is the product of its diagonal elements,

, A - \lambda I,  = \begin
    2 - \lambda & 0 & 0 & 0 \\
    1 & 2- \lambda & 0 & 0 \\
    0 & 1 & 3- \lambda & 0  \\
    0 & 0 & 1 & 3- \lambda
  \end =
  (2 - \lambda)^2(3 - \lambda)^2.

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

\begin 0 & 1 & -1 & 1 \end^\textsf

and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector

\begin 0 & 0 & 0 & 1 \end^\textsf

. 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,

, v_, ^2 = \frac,

where

M_j

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, \begi ...

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 ...

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 operators 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

D f(t) = \lambda f(t)

The functions that satisfy this equation are eigenvectors of ''D'' and are commonly called eigenfunctions.

Derivative operator example

Consider the derivative operator

\tfrac

with eigenvalue equation

\fracf(t) = \lambda f(t).

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 ...

f(t) = f(0)e^,

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

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

''K'' of scalars, and let ''T'' be a linear transformation mapping ''V'' into ''V'',

T:V \to 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

E = \left\,

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,

\begin
  T(\mathbf + \mathbf) &= T(\mathbf) + T(\mathbf),\\
        T(\alpha \mathbf) &= \alpha T(\mathbf),
\end

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

\begin
  T(\mathbf + \mathbf) &= \lambda (\mathbf + \mathbf),\\
        T(\alpha \mathbf) &= \lambda (\alpha \mathbf).
\end

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 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 * Mo ...

. 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 :

x_t = a_1 x_ + a_2 x_ + \cdots + a_k x_.

The solution of this equation for ''x'' in terms of ''t'' is found by using its characteristic equation :

\lambda^k - a_1\lambda^ - a_2\lambda^ - \cdots - a_\lambda-a_k = 0,

which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the ''k'' – 1 equations

x_ = x_,\  \dots,\  x_ = x_,

giving a ''k''-dimensional system of the first order in the stacked variable vector

\begin x_t & \cdots & x_ \end

in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives ''k'' characteristic roots

\lambda_1,\, \ldots,\, \lambda_k,

for use in the solution equation :

x_t = c_1\lambda_1^t + \cdots + c_k\lambda_k^t.

A similar procedure is used for solving a

of the form :

\frac + a_\frac + \cdots + a_1\frac + a_0 x = 0.

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

A

can be determined by finding the roots of the characteristic polynomial. This is easy for

2 \times 2

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

, which for an

n \times n

matrix is a sum of

n!

different products. Explicit algebraic formulas for the roots of a polynomial exist only if the degree

n

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

n

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

n

.) 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

A = \begin 4 & 1\\ 6 & 3\end

we can find its eigenvectors by solving the equation

A v = 6 v

, that is

\begin 4 & 1\\ 6 & 3\end\beginx \\y\end = 6 \cdot \beginx \\y\end

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 ...

\left\{ \begin{aligned} 4x + y &= 6x \\ 6x + 3y &= 6y\end{aligned} \right.

that is

\left\{ \begin{aligned} -2x + y &= 0 \\ 6x - 3y &= 0\end{aligned} \right.

Both equations reduce to the single linear equation

y=2x

. Therefore, any vector of the form

\begin{bmatrix} a & 2a \end{bmatrix}^\textsf{T}

, for any nonzero real number

a

, is an eigenvector of

A

with eigenvalue

\lambda = 6

. The matrix

A

above has another eigenvalue

\lambda=1

. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of

3x+y=0

, that is, any vector of the form

\begin{bmatrix} b & -3b \end{bmatrix}^\textsf{T}

, for any nonzero real number

b

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

\mathbf{v}

is (a good approximation of) an eigenvector of

A

, then the corresponding eigenvalue can be computed as :

\lambda = \frac{\mathbf{v}^* A\mathbf{v{\mathbf{v}^* \mathbf{v

where

\mathbf{v}^*

denotes the

\mathbf{v}

Modern methods

Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the

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 transformati ...

with the LU decomposition results in an algorithm with better convergence than the QR algorithm. For large

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 , t ...

, - ! scope="row" , Illustration , Homothety in two dim

, - style="vertical-align:top" ! scope="row" , Matrix ,

\begin{bmatrix}k & 0\\ 0 & k\end{bmatrix}

\begin{bmatrix}k_1 & 0\\ 0 & k_2\end{bmatrix}

\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}

\begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix}

\begin{bmatrix}\cosh\varphi & \sinh\varphi\\ \sinh\varphi & \cosh\varphi\end{bmatrix}

, - ! scope="row" , Characteristic
polynomial ,

\ (\lambda - k)^2

(\lambda - k_1)(\lambda - k_2)

\lambda^2 - 2\cos(\theta)\lambda + 1

\ (\lambda - 1)^2

\lambda^2 - 2\cosh(\varphi)\lambda + 1

, - ! scope="row" , Eigenvalues,

\lambda_i

\lambda_1 = \lambda_2 = k

\begin{align}\lambda_1 &= k_1 \\ \lambda_2 &= k_2\end{align}

\begin{align}\lambda_1 &= e^{i\theta} \\ &= \cos\theta + i\sin\theta \\ \lambda_2 &= e^{-i\theta} \\ &= \cos\theta - i\sin\theta \end{align}

\lambda_1 = \lambda_2 = 1

\begin{align}\lambda_1 &= e^\varphi \\ &= \cosh\varphi + \sinh\varphi \\ \lambda_2 &= e^{-\varphi} \\ &= \cosh\varphi - \sinh\varphi \end{align}

, - ! scope="row" , Algebraic ,

\mu_i = \mu(\lambda_i)

\mu_1 = 2

\begin{align}\mu_1 &= 1 \\ \mu_2 &= 1 \end{align}

\begin{align}\mu_1 &= 1 \\ \mu_2 &= 1 \end{align}

\mu_1 = 2

\begin{align}\mu_1 &= 1 \\ \mu_2 &= 1 \end{align}

, - ! scope="row" , Geometric ,

\gamma_i = \gamma(\lambda_i)

\gamma_1 = 2

\begin{align}\gamma_1 &= 1 \\ \gamma_2 &= 1 \end{align}

\begin{align}\gamma_1 &= 1 \\ \gamma_2 &= 1 \end{align}

\gamma_1 = 1

\begin{align}\gamma_1 &= 1 \\ \gamma_2 &= 1 \end{align}

, - ! scope="row" , Eigenvectors , All nonzero vectors ,

\begin{align}
  \mathbf u_1 &= \begin{bmatrix} 1\\ 0\end{bmatrix} \\
  \mathbf u_2 &= \begin{bmatrix} 0\\ 1\end{bmatrix}
\end{align}

\begin{align}
  \mathbf u_1 &= \begin{bmatrix} 1\\ -i\end{bmatrix} \\
  \mathbf u_2 &= \begin{bmatrix} 1\\ +i\end{bmatrix}
\end{align}

\mathbf u_1 = \begin{bmatrix} 1\\ 0 \end{bmatrix}

\begin{align}
  \mathbf u_1 &= \begin{bmatrix} 1\\  1\end{bmatrix} \\
  \mathbf u_2 &= \begin{bmatrix} 1\\ -1\end{bmatrix}
\end{align}

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 ...

D = -4(\sin\theta)^2

, 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,

\cos\theta \pm i\sin\theta

; 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

An example of an eigenvalue equation where the transformation

T

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 ...

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, ...

: :

H\psi_E = E\psi_E \,

where

H

, 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 and

\psi_E

, 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

E

, 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

\psi_E

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

\psi_E

and

H

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 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

, \Psi_E\rangle

. In this notation, the Schrödinger equation is: :

H, \Psi_E\rangle = E, \Psi_E\rangle

where

, \Psi_E\rangle

is an eigenstate of

H

and

E

represents the eigenvalue.

H

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 its ...

, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above

H, \Psi_E\rangle

is understood to be the vector obtained by application of the transformation

H

, \Psi_E\rangle

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 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

\mathbf{t}

. The eigenvectors of the transmission operator

\mathbf{t}^\dagger\mathbf{t}

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,

\tau

, of

\mathbf{t}^\dagger\mathbf{t}

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

\tau_\max = 1

and

\tau_\min = 0

. Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.

Molecular orbitals

, 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 aft ...

. 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 ...

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 mat ...

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

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

\mathbf v_1, \mathbf v_2, \mathbf v_3

by their eigenvalues

E_1 \geq E_2 \geq E_3

;

\mathbf v_1

then is the primary orientation/dip of clast,

\mathbf v_2

is the secondary and

\mathbf v_3

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

E_1

E_2

, and

E_3

are dictated by the nature of the sediment's fabric. If

E_1 = E_2 = E_3

, the fabric is said to be isotropic. If

E_1 = E_2 > E_3

, the fabric is said to be planar. If

E_1 > E_2 > E_3

, the fabric is said to be linear.

Principal component analysis

The

of a

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 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 numbe ...

equal to one). For the covariance or correlation matrix, the eigenvectors correspond to principal components and the eigenvalues to the variance explained by the principal components. Principal component analysis of the correlation matrix provides an

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 ...

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 ...

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

Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. 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

m\ddot{x} + kx = 0

m\ddot{x} = -kx

that is, acceleration is proportional to position (i.e., we expect

x

to be sinusoidal in time). In

n

dimensions,

m

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

k

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

kx = \omega^2 mx

where

\omega^2

is the eigenvalue and

\omega

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

k

alone. Furthermore, damped vibration, governed by

m\ddot{x} + c\dot{x} + kx = 0

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 tha ...

\left(\omega^2 m + \omega c + k\right)x = 0.

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

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

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 Ale ...

s; this is an example of

. 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 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

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

. 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

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

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 ...

A

, or (increasingly) of the graph's Laplacian matrix 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 or a discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertice ...

, which is either

D - A

(sometimes called the ''combinatorial Laplacian'') or

I - D^{-1/2}A D^{-1/2}

(sometimes called the ''normalized Laplacian''), where

D

is a diagonal matrix with

D_{ii}

equal to the degree of vertex

v_i

, and in

D^{-1/2}

, the

i

th diagonal entry is

1/\sqrt{\deg(v_i)}

. The

k

th principal eigenvector of a graph is defined as either the eigenvector corresponding to the

k

th largest or

k

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. ...

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

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 (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided as ...

. Other methods are also available for clustering.

Basic reproduction number

The basic reproduction number (

R_0

) 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

R_0

is the average number of people that one typical infectious person will infect. The generation time of an infection is the time,

t_G

, 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

t_G

has passed.

R_0

is then the largest eigenvalue of the next generation matrix.

Notes

Citations

Sources

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

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 Calculator
from Symbolab (Click on the bottom right button of the 2x12 grid to select a matrix size. Select an

n \times n

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, 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 BiCGST ...

{{DEFAULTSORT:Eigenvalues And Eigenvectors Abstract algebra Linear algebra Mathematical physics Matrix theory Singular value decomposition

Formal definition

Overview

History

Eigenvalues and eigenvectors of matrices

Eigenvalues and the characteristic polynomial

Algebraic multiplicity

Eigenspaces, geometric multiplicity, and the eigenbasis for matrices

Additional properties of eigenvalues

Left and right eigenvectors

Diagonalization and the eigendecomposition

Variational characterization

Matrix examples

Two-dimensional matrix example

Three-dimensional matrix example

Three-dimensional matrix example with complex eigenvalues

Diagonal matrix example

Triangular matrix example

Matrix with repeated eigenvalues example

Eigenvector-eigenvalue identity

Eigenvalues and eigenfunctions of differential operators

Derivative operator example

General definition

Eigenspaces, geometric multiplicity, and the eigenbasis

Spectral theory

Associative algebras and representation theory

Dynamic equations

Calculation

Classical method

Eigenvalues

Eigenvectors

Simple iterative methods

Modern methods

Applications

Eigenvalues of geometric transformations

Schrödinger equation

Wave transport

Molecular orbitals

Geology and glaciology

Principal component analysis

Vibration analysis

Eigenfaces

Tensor of moment of inertia

Stress tensor

Graphs

Basic reproduction number

See also

Notes

Citations

Sources

Further reading

External links

Theory