1. Definitions and properties of f (A)

1.1 Examples of matrix functions

There are many applications involving matrix functions. We often manipulate matrix functions without realizing it. For example, when it exists, the inverse A ^-1 of A corresponds to the function such that f (x) = 1/x. Solving the linear system Ax = b means implicitly applying f (A) = A ^-1 to the vector b.

Another simple example is the solution of systems of linear differential equations involving the exponential function. We want to calculate y, solution of the equation