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.