Matrix function

In mathematics, a matrix function is a function which maps a matrix to another matrix.

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.

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 ${\displaystyle |x|<r}$, then the corresponding matrix series will converge for matrix argument A if ${\displaystyle \|A\|<r}$ for some matrix norm ${\displaystyle \|\cdot \|}$ which satisfies ${\displaystyle \|AB\|\leq \|A\|\cdot \|B\|}$.

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