In
mathematics, every
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
can be used for defining a matrix function that maps
square matrices with complex entries to square matrices of the same size.
This is used for defining the
exponential of a matrix
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 giv ...
, which is involved in the
closed-form solution of systems of
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ ...
s.
Extending scalar function to matrix functions
There are several techniques for lifting a real function to 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 ofte ...
function such that interesting properties are maintained. All of the following techniques yield the same matrix function, but the domains on which the function is defined may differ.
Power series
If the
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
has the
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
then a matrix function
can be defined by substituting by 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 ofte ...
: powers become
matrix powers, additions become matrix sums and multiplications by coefficients become
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 ...
s. If the series converges for
, then the corresponding matrix series converges for matrices such that
for some
matrix norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ...
that satisfies
.
Diagonalizable matrices
A square matrix is
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 ...
, if there is an
invertible 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 ...
such that
is a
diagonal matrix
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 ...
, that is, has the shape
As
it is natural to set
It can be verified that the matrix does not depend on a particular choice of .
For example, suppose one is seeking
for
One has
for
Application of the formula then simply yields
Likewise,
Jordan decomposition
All complex matrices, whether they are diagonalizable or not, have a
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to ...
, where the matrix ''J'' consists of
Jordan blocks. Consider these blocks separately and apply the power series to a Jordan block:
This definition can be used to extend the domain of the matrix function
beyond the set of matrices with
spectral radius
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spect ...
smaller than the radius of convergence of the power series.
Note that there is also a connection to
divided differences.
A related notion is the
Jordan–Chevalley decomposition which expresses a matrix as a sum of a diagonalizable and a nilpotent part.
Hermitian matrices
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 -t ...
has all real eigenvalues and can always be diagonalized by a
unitary matrix
In linear algebra, a Complex number, complex Matrix (mathematics), square matrix is unitary if its conjugate transpose is also its Invertible matrix, inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, esp ...
P, according to the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful b ...
.
In this case, the Jordan definition is natural. Moreover, this definition allows one to extend standard inequalities for
real functions:
If
for all eigenvalues of
, then
.
(As a convention,
is a
positive-semidefinite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...
.)
The proof follows directly from the definition.
Cauchy integral
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
from
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
can also be used to generalize scalar functions to matrix functions. Cauchy's integral formula states that for any
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
defined on a set , one has
where is a closed simple curve inside the domain enclosing .
Now, replace by a matrix and consider a path inside that encloses all
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 denot ...
s of . One possibility to achieve this is to let be a circle around the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
with
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
larger than for an arbitrary
matrix norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ...
. Then, is definable by
This integral can readily be evaluated numerically using the
trapezium rule, which
converges exponentially in this case. That means that the
precision of the result doubles when the number of nodes is doubled. In routine cases, this is bypassed by
Sylvester's formula In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function of a matrix as a polynomial in , in terms of the eigenvalues and eigenvectors ...
.
This idea applied to
bounded linear operators on a
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 ve ...
, which can be seen as infinite matrices, leads to the
holomorphic functional calculus.
Matrix perturbations
The above Taylor power series allows the scalar
to be replaced by the matrix. This is not true in general when expanding in terms of
about
unless
. A counterexample is
, which has a finite length Taylor series. We compute this in two ways,
* Distributive law:
* Using scalar Taylor expansion for
and replacing scalars with matrices at the end:
The scalar expression assumes commutativity while the matrix expression does not, and thus they cannot be equated directly unless
. For some ''f''(''x'') this can be dealt with using the same method as scalar Taylor series. For example,
. If
exists then
. The expansion of the first term then follows the power series given above,
The convergence criteria of the power series then apply, requiring
to be sufficiently small under the appropriate matrix norm. For more general problems, which cannot be rewritten in such a way that the two matrices commute, the ordering of matrix products produced by repeated application of the Leibniz rule must be tracked.
Arbitrary function of a 2×2 matrix
An arbitrary function ''f''(''A'') of a 2×2 matrix A has its
Sylvester's formula In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function of a matrix as a polynomial in , in terms of the eigenvalues and eigenvectors ...
simplify to
where
are the eigenvalues of its characteristic equation, , and are given by
Examples
*
Matrix polynomial
*
Matrix root
*
Matrix logarithm
*
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 giv ...
*
Matrix sign function
Classes of matrix functions
Using the
semidefinite ordering (
is
positive-semidefinite and
is
positive definite), some
of the classes of scalar functions can be extended to matrix functions of
Hermitian matrices.
Operator monotone
A function is called operator monotone if and only if
for all self-adjoint matrices with spectra in the domain of . This is analogous to
monotone function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
in the scalar case.
Operator concave/convex
A function is called operator concave if and only if
for all self-adjoint matrices with spectra in the domain of and
Examples
The matrix log is both operator monotone and operator concave. The matrix square is operator convex. The matrix exponential is none of these. Loewner's theorem states that a function on an ''open'' interval is operator monotone if and only if it has an analytic extension to the upper and lower complex half planes so that the upper half plane is mapped to itself.
[
]
See also
* Algebraic Riccati equation
*Sylvester's formula In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function of a matrix as a polynomial in , in terms of the eigenvalues and eigenvectors ...
* Loewner order
* Matrix calculus
* Trace inequalities
* Trigonometric functions of matrices
Notes
References
* {{cite book, last1=Higham, first1=Nicholas J., title=Functions of matrices theory and computation, date=2008, publisher=Society for Industrial and Applied Mathematics, location=Philadelphia, author-link=Nicholas_Higham, isbn=9780898717778
Matrix theory
Mathematical physics