Diagonalizability properties

Question

Say matrices X and Y are both diagonalisable and they have the same set of eigenvalues. Each of the eigenvalues have the same algebraic and geometry multiplicities. From here, I can see that the Tr(X) and Tr(Y) are equal and same goes with their determinants since their eigenvalues are the same. But what does it say about their eigenspaces? Are the eigenspaces of both matrices X and Y similar with respect to each eigenvalue by that i mean for every E_$\lambda$, E_$\lambda$(X)=E_$\lambda$(Y) . Are these equal? The det(X-$\lambda$I) and det(Y - $\lambda$I) are both equal as well right? It seems pretty simple but i cant seem to get it.