# When do two different matrices have the same eigenvalues and eigenvectors?

What are the conditions on two different matrices having the same eigenvalues and eigenvectors?

For example, the $2\times 2$ matrices:
$$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \; , \; \begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix}$$
have the same eigenvalue (zero) and the same eigenvector $[1\; 0]^T$ (or a nonzero multiple thereof).
If you mean "What does it mean for $A$ and $B$ both to have the same set of eigenvalues, occurring in each care with the same multiplicity", then the answer is that $A$ and $B$ are similar: $B=P^{-1}AP$ for some non-singular $P$.