Is the following a good example?
$P=\begin{bmatrix}1&1\\1&1\end{bmatrix}$
then, multiply the first row with 1 and subtract the first row from the second row, we can get:
$P'=\begin{bmatrix}1&1\\0&0\end{bmatrix}$
Eigenvalues P= 0,2 Eigenvalues P'=0,1
I wonder whether this can be a good example for the problem.