## Functions of Matrices

Given a square matrix
$A$
we can find
$f(A)$
for a wide variety of functions
$f(A)$
. The result may be a number, or another matrix.
Example: Let
$A= \left( \begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array} \right)$
and
$f(x)=x^2-3x-2$

Then
$f(A)=A^2-3A-2I$
where
$I$
is the identity 2 x 2 matrix.
\begin{aligned} f(A) &= A^2-3A+2I \\ &= \left( \begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array} \right) \left( \begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array} \right) - 3 \left( \begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array} \right) +2 \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \\ &= \left( \begin{array}{cc} 3 & 9 \\ 3 & 9 \end{array} \right) \end{aligned}

Possible functions
$f$
include the trigonometric functions, the exponential and log functions, and in fact any function which can be expanded in a Taylor or Mclaurin power series.