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.
It can be verified that the matrix f(A) does not depend on a particular choice of P. For example, suppose one is seeking
Application of the formula then simply yields
All complex 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.
Moreover, this definition allows one to extend standard inequalities for real functions: If
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 simple 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 ‖·‖.
This integral can readily be evaluated numerically using the trapezium rule, which converges exponentially in this case.
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.
This is not true in general when expanding in terms of
, which has a finite length Taylor series.
We compute this in two ways, 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.
The convergence criteria of the power series then apply, requiring
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.
An arbitrary function f(A) of a 2×2 matrix A has its Sylvester's formula simplify to
is positive definite), some of the classes of scalar functions can be extended to matrix functions of Hermitian matrices.
for all self-adjoint matrices A,H with spectra in the domain of f. This is analogous to monotone function in the scalar case.
This definition is analogous to a concave scalar function.
An operator convex function can be defined be switching
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.