For a matrix
\begin{array}{cc} a & 0 \\ 0 & b \\ \end{array}
and \begin{array}{cc} b & 0 \\ 0 & a \\ \end{array} how do we prove if they are similar? They have the same characteristic polynomial, and same rank. Are there other attributes of similar matrices that might confirm their similarity? If not, how can we prove they aren't similar? I wasn't sure how to use the A = $P^{-1}$BP characteristic of similar matrices.
Yes, they are similar. Just take $P=\left[\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right]$ and check that $\left[\begin{smallmatrix}b&0\\0&a\end{smallmatrix}\right]=P.\left[\begin{smallmatrix}a&0\\0&b\end{smallmatrix}\right].P^{-1}$.