I have the two matrices:
$\begin{pmatrix}1&-4&-2\\ 0&1&0\\ 0&4&3\end{pmatrix}$ and $\begin{pmatrix}3&0&0\\ \:0&1&1\\ \:0&0&1\end{pmatrix}$
I know they have the same trace and determinant but I know that isn't enough to prove they are similar… what are the next steps I should take?
Is proving they have the same eigenvalues enough to show they are similar?
Best Answer
The first matrix is diagonalizable, namely $P^{-1}AP={\rm diag}(1,1,3)$ with $$ P=\begin{pmatrix} -3 & 2 & -2\cr -2 & 1 & 0 \cr 4 & -2 & 2 \end{pmatrix}. $$ The second matrix is not diagonalizable, so they are in fact not similar.