\[A\]
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.\[\begin{equation} \begin{aligned} f(A) &= A^2-3A+2I \\ &= \left( \begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array} \right) \left( \begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array} \right) - 3 \left( \begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array} \right) +2 \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \\ &= \left( \begin{array}{cc} 3 & 9 \\ 3 & 9 \end{array} \right) \end{aligned} \end{equation}\]
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.