In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
and applications, matrix analysis is the study 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 ...
and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as
matrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered addition for matrices, such as the direct sum and the Kroneck ...
,
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
and operations derived from these), functions of matrices (such as
matrix exponentiation
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives ...
and
matrix logarithm
In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exp ...
, and even
sines and cosines
Sines () is a city and a municipality in Portugal. The municipality, divided into two parishes, has around 14,214 inhabitants (2021) in an area of . Sines holds an important oil refinery and several petrochemical industries. It is also a popular ...
etc. of matrices), and the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of matrices (
eigendecomposition of a matrix
In linear algebra, eigendecomposition is the Matrix factorization, factorization of a matrix (mathematics), matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrix, di ...
,
eigenvalue perturbation
In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system Ax=\lambda x that is perturbed from one with known eigenvectors and eigenvalues A_0 x=\lambda_0x_0 . This is useful for studyi ...
theory).
[
]
Matrix spaces
The set of all ''m'' × ''n'' matrices 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 ...
''F'' denoted in this article ''M''
''mn''(''F'') form a
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 ...
. Examples of ''F'' include the set of
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
, the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s
, and set of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s
. The spaces ''M''
''mn''(''F'') and ''M''
''pq''(''F'') are different spaces if ''m'' and ''p'' are unequal, and if ''n'' and ''q'' are unequal; for instance ''M''
32(''F'') ≠ ''M''
23(''F''). Two ''m'' × ''n'' matrices A and B in ''M''
''mn''(''F'') can be added together to form another matrix in the space ''M''
''mn''(''F''):
:
and multiplied by a ''α'' in ''F'', to obtain another matrix in ''M''
''mn''(''F''):
:
Combining these two properties, a
linear combination of matrices A and B are in ''M''
''mn''(''F'') is another matrix in ''M''
''mn''(''F''):
:
where ''α'' and ''β'' are numbers in ''F''.
Any matrix can be expressed as a linear combination of basis matrices, which play the role of the
basis vector
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s for the matrix space. For example, for the set of 2 × 2 matrices over the field of real numbers,
, one legitimate basis set of matrices is:
:
because any 2 × 2 matrix can be expressed as:
:
where ''a'', ''b'', ''c'',''d'' are all real numbers. This idea applies to other fields and matrices of higher dimensions.
Determinants
The determinant of 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 ...
is an important property. The determinant indicates if a matrix is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
(i.e. the
inverse of a matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
exists when the determinant is nonzero). Determinants are used for finding eigenvalues of matrices (see below), and for solving 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 ...
(see
Cramer's rule).
Eigenvalues and eigenvectors of matrices
Definitions
An ''n'' × ''n'' matrix A has eigenvectors x and eigenvalues ''λ'' defined by the relation:
:
In words, the
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
of A followed by an eigenvector x (here an ''n''-dimensional
column matrix), is the same as multiplying the eigenvector by the eigenvalue. For an ''n'' × ''n'' matrix, there are ''n'' eigenvalues. The eigenvalues are the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of 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 ...
:
:
where I is the ''n'' × ''n''
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 ...
.
Roots of polynomials, in this context the eigenvalues, can all be different, or some may be equal (in which case eigenvalue has
multiplicity
Multiplicity may refer to: In science and the humanities
* Multiplicity (mathematics), the number of times an element is repeated in a multiset
* Multiplicity (philosophy), a philosophical concept
* Multiplicity (psychology), having or using mult ...
, the number of times an eigenvalue occurs). After solving for the eigenvalues, the eigenvectors corresponding to the eigenvalues can be found by the defining equation.
Perturbations of eigenvalues
Matrix similarity
Two ''n'' × ''n'' matrices A and B are similar if they are related by a similarity transformation:
:
The matrix P is called a similarity matrix, and is necessarily
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 ...
.
Unitary similarity
Canonical forms
Row echelon form
Jordan normal form
Weyr canonical form
Frobenius normal form
Triangular factorization
LU decomposition
LU decomposition splits a matrix into a matrix product of an upper
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 ...
and a lower triangle matrix.
Matrix norms
Since matrices form vector spaces, one can form axioms (analogous to those of vectors) to define a "size" of a particular matrix. The norm of a matrix is a positive real number.
Definition and axioms
For all matrices A and B in ''M''
''mn''(''F''), and all numbers ''α'' in ''F'', a matrix norm, delimited by double vertical bars , , ... , , , fulfills:
[Some authors, e.g. Horn and Johnson, use triple vertical bars instead of double: , , , A, , , .]
*
Nonnegative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
:
::
:with equality only for A = 0, the
zero matrix In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followed ...
.
*
Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by ...
:
::
*The
triangular inequality:
::
Frobenius norm
The Frobenius norm is analogous to the
dot 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 ...
of Euclidean vectors; multiply matrix elements entry-wise, add up the results, then take the positive
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
:
:
It is defined for matrices of any dimension (i.e. no restriction to square matrices).
Positive definite and semidefinite matrices
Functions
Matrix elements are not restricted to constant numbers, they can be
mathematical variable
In mathematics, a variable (from Latin language, Latin ''wikt:variabilis, variabilis'', "changeable") is a Mathematical symbol, symbol that represents a mathematical object. A variable may represent a number, a Vector (mathematics), vector, a Mat ...
s.
Functions of matrices
A functions of a matrix takes in a matrix, and return something else (a number, vector, matrix, etc...).
Matrix-valued functions
A matrix valued function takes in something (a number, vector, matrix, etc...) and returns a matrix.
See also
Other branches of analysis
*
Mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
*
Tensor analysis
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
*
Matrix calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a ...
*
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
Other concepts of linear algebra
*
Tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
*
Spectrum of an operator
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number \lambda is ...
*
Matrix geometrical series
In mathematics, a matrix polynomial is a polynomial with square matrix, square matrices as variables. Given an ordinary, scalar-valued polynomial
: P(x) = \sum_^n =a_0 + a_1 x+ a_2 x^2 + \cdots + a_n x^n,
this polynomial evaluated at a matrix ' ...
Types of matrix
*
Orthogonal matrix
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 ...
,
unitary matrix
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is ...
*
Symmetric matrix
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 ...
,
antisymmetric matrix
Antisymmetric or skew-symmetric may refer to:
* Antisymmetry in linguistics
* Antisymmetric relation in mathematics
* Skew-symmetric graph
* Self-complementary graph
In mathematics, especially linear algebra, and in theoretical physics, the adject ...
*
Stochastic matrix
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ...
Matrix functions
*
Matrix polynomial
In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial
: P(x) = \sum_^n =a_0 + a_1 x+ a_2 x^2 + \cdots + a_n x^n,
this polynomial evaluated at a matrix ''A'' is
:P(A) = ...
*
Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives ...
Footnotes
References
Notes
Further reading
*
*
*
*{{cite book, title=Computational Matrix Analysis, author=Alan J. Laub, year=2012, publisher=SIAM, isbn=978-161-197-221-4, url=https://books.google.com/books?id=RJBZBuHpVjEC&q=Matrix+Analysis
Linear algebra
Matrices
Numerical analysis