## Similar or Conjugate Matrices

Matrices
$A$
and
$B$
are similar if there exists an invertible matrix
$P$
such that
$B=P^{-1}AP$

The matrices
$\left( \begin{array}{cc} 4 & 3 \\ 2 & 1 \end{array} \right)$
and
$\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)$
are similar with
$P= \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$

Note that
$\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)^{-1} =\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$
so that
$P$
is self inverse.
$\left( \begin{array}{cc} 4 & 3 \\ 2 & 1 \end{array} \right) = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)^{-1} \left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right) \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$

$B$
is also said to be conjugate to
$B$
and vice versa.
Only square matrices can be similar.