additional hypotheses to conclude that two matrices with the same eigenvalues are actually equal

Question

If two matrices have the same eigenvalues, what are some minimal additional hypotheses one could add to conclude they are equal? (e.g. they are symmetric, they are positive definite, etc.)

Answer 1

Note that two matrices are equal $\iff$ the corresponding entries are equals.

We define two matrices A and B similar $\iff$ they represent the same linear operator with respect to (possibly) different bases, that is

$$B = P^{-1} A P$$

in this case A and B share the same eigenvalues and also others properties.

Answer 2

If a matrix is diagonalizable, M = $S^{-1}DS$, then knowing the eigenvalues gives you D. Knowing the eigenvectors gives you S. So those two pieces of information would be significant. If the matrix is not diagonlizable, then knowing the Jordan canonical form and the transformations to get there would be equivalent to knowing the matrix. Thus, anything equivalent to all that information would also be sufficient. So if you have the Jordan canonical form, and the nature of the spaces associated with each eigenvalue, that would be sufficient.