There are many ways to define a** matrix function** (or *matrix-valued function*). One way is that the function has the form of a matrix, but instead of numbers inside the matrix there are functions. For example, **A**(t) is a 2 x 2 matrix function:

The function is defined only for t values that define all 4 of the elements in this example matrix.

Another way to define a matrix function is simply as a function that calculates a (usually square) matrix **A**; To put that another way, it’s a function that maps one matrix to another. Some authors define matrix functions on the complex plane ℂ. For example Higham (2008) highlights the definition that it involves a scalar function f and a matrix **A** ∈ ℂ^{n x n}; the combination calculates a matrix the same dimensions as **A**.

An *entire* matrix function calculates a matrix and the action of that matrix on a vector (Schilders, et al., 2008). Matrix functions play an important role in control theory and the theory of differential equations.

## Matrix Function Examples

- The powers
**A**^{p}, where**A**is an*n*x*n*matrix, are all matrix functions (Abadir & Magnus, 2005). - The identity function f(t) = t is a monotone matrix function for every real interval (Johnson, 1990).

## Matrix Functions by Infinite Series

A matrix function for square matrices **A** can be defined by the infinite series

as long as there is a finite limit (Peterson & Pedersen, 2005).

## References

Abadir, K. & Magnus, J. (2005). Matrix Algebra. Cambridge University Press.

Higham, N. (2008). Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics). Society for Industrial and Applied Mathematics.

Johnson, C. (1990). Matrix Theory and Applications. Proc Held Phoenix, Jan 10-11, 1989 (Proceedings of Symposia in Applied Mathematics). American Mathematical Society.

Peterson, K. & Pedersen, M. (2005). The Matrix Cookbook. Retrieved January 21, 2021 from: https://www.ics.uci.edu/~welling/teaching/KernelsICS273B/MatrixCookBook.pdf

Schilders, W. et al., (2008). Model Order Reduction: Theory, Research Aspects and Applications. Springer Berlin Heidelberg.