Analytic function of a matrix

A matrix function usually denotes a function which maps a matrix to a matrix.

Extending scalar functions to matrix functions

There are several techniques for lifting a real function to a matrix function such that some interesting properties are maintained.

Power series

If the real function $f$ has the Taylor expansion

$f(x)=f(0)+f'(0)\cdot x+f''(0)\cdot {\frac {x^{2}}{2}}+\dots$

you can substitute $x$ 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 $|x|<r$ then the corresponding matrix series will converge for matrix argument $M$ if there is a matrix norm $||\cdot ||$ with $||M||<r$.