If $AB = BA$, how are the eigenvalues of $A$ related to those of $B$?

Question

Suppose $A$ and $B$ are two symmetric real square matrices, such that $AB = BA$. If $\lambda_{1},\dotsc,\lambda_{n}$ are the eigenvalues of $A$, can we find the eigenvalues of $B$?

Answer 1

The eigenvalues of $A$ and $B$ can be any values. Just let $A$ and $B$ be diagonal matrices.

Answer 2

What can you conclude if $A$ is the $0$ matrix?

Answer 3

No. For simplicity, just use $A$ and $B$ diagonal matrices, so the diagonals are the eigenvalues. Then, with $\mu_i$ being the eigenvalues of $B$, you get $AB=BA$, since they are both diagonals, with $\lambda_i\mu_i$ on the diagonal. The above holds for any $\mu_i$