If $A$ and $B$ are similar matrices then every eigenvector of A is an eigenvector of B, true or false? I know if $A$ and $B$ are similar matrices, then there exists an invertible matrix $P$ such that $B = P^{-1} A P$. Also,
1) the trace of similar matrices are equal
2) the determinant and characteristic polynomial of similar matrices are equal.
I would like a generalized proof of my question. Thanks.
The matrices $$ A = \left(\begin{array}{cc} 0 & 2\\ 2 & 0\end{array}\right)\qquad\text{and}\qquad B= \left(\begin{array}{cc} 2 & 0\\ 0 & -2\end{array}\right) $$ are similar (as you should check). The vector $(1,-1)^T$ is an eigenvector of $A$, but it is not an eigenvector of $B$ (you should check this as well).