Matrix function

In mathematics, a function maps an input value to an output value. In the case of a matrix function, the input and the output values are matrices. One example of a matrix function occurs with the Algebraic Riccati equation, which is used to solve certain optimal control problems. 
Matrix functions are special functions made by matrices.
Definitions.
Most functions like formula_1 are defined as a solution of a differential equation. But matrix functions will use a different way.
Suppose formula_2 is a complex number and formula_3 is a square matrix. If you have a polynomial:
then it is reasonable to define
Let's use this idea. When you have
then you can introduce
For example, the matrix version of the exponential function and the trigonometric functions are defined as follows:
Importance.
Matrix functions are used at numerical methods for ordinary differential equations and statistics. This is why numerical analysts are studying how to compute them. For example, the following functions are studied: