Isomorphism in the case of real symmetric matrices

Question

I'm looking for clarification of the following problem: it's known that determining which graphs are uniquely determined by their spectra is in general a very hard problem. But what about more general problem - real symmetric square matrices? 1) Do we have the same problem when two cospectral matrices can be non-isomorphic? And a related question: 2) is there any simple method to test isomorphism of two arbitrary real symmetric square matrices? Many Thanks