In mathematics, a matrix function is a function which maps a matrix to another matrix.
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Extending scalar function to matrix functions
There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained. All of the following techniques yield the same matrix function, but the domains on which the function are defined may differ.
Power series
If the real function f has the Taylor expansion
then a matrix function can be defined by substituting x by a matrix: the powers become matrix powers, the additions become matrix sums and the multiplications become scaling operations. If the real series converges for
Diagonalizable matrices
If the matrix A is diagonalizable, the problem may be reduced to an array of the function on each eigenvalue. This is to say we can find a matrix P and a diagonal matrix D such that
where
For example, suppose one is seeking A! = Γ(A+1) for
One has
for
Application of the formula then simply yields
Likewise,
Jordan decomposition
All matrices, whether they are diagonalizable or not, have a Jordan normal form
This definition can be used to extend the domain of the matrix function beyond the set of matrices with spectral radius 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 has all real eigenvalues and can always be diagonalized by a unitary matrix P, according to the spectral theorem. In this case, the Jordan definition is natural. Moreover, this definition allows one to extend standard inequalities for real functions:
If
Cauchy integral
Cauchy's integral formula from complex analysis can also be used to generalize scalar functions to matrix functions. Cauchy's integral formula states that for any analytic function f defined on a set D ⊂ ℂ, one has
where C is a closed curve inside the domain D enclosing x.
Now, replace x by a matrix A and consider a path C inside D that encloses all eigenvalues of A. One possibility to achieve this is to let C be a circle around the origin with radius larger than ‖A‖ for an arbitrary matrix norm ‖•‖. Then, f(A) 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.
This idea applied to bounded linear operators on a Banach space, which can be seen as infinite matrices, leads to the holomorphic functional calculus.
Matrix perturbations
The above Taylor power series allows the scalar
The scalar expression assumes commutativity while the matrix expression does not and thus they cannot be equated directly unless
The convergence criteria of the power series then apply, requiring
Examples
Classes of matrix functions
Using the semidefinite ordering (
Operator monotone
A function
Operator concave/convex
A function
for all self-adjoint matrices
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.