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Concept
Analytic function of a matrix
Summary
In mathematics, every analytic function 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, which is involved in the closed-form solution of systems of linear differential equations.
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 is defined may differ.
If the analytic function f has the Taylor expansion
then a matrix function can be defined by substituting x by a square matrix: powers become matrix powers, additions become matrix sums and multiplications by coefficients become scalar multiplications. If the series converges for , then the corresponding matrix series converges for matrices A such that for some matrix norm that satisfies .
A square matrix A is diagonalizable, if there is an invertible matrix P such that is a diagonal matrix, that is, D has the shape
As it is natural to set
It can be verified that the matrix f(A) does not depend on a particular choice of P.
For example, suppose one is seeking for
One has
for
Application of the formula then simply yields
Likewise,
Jordan normal form
All complex matrices, whether they are diagonalizable or not, have a Jordan normal form , 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 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.
A Hermitian matrix has all real eigenvalues and can always be diagonalized by a unitary matrix P, according to the spectral theorem.
Official source
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