Common eigenvectors of two matrices

Questions
1

I know that using Mathematica you can obtain the eigenvalues and eigenvectors of a matrix. I was wondering if using Mathematica you can obtain the common eigenvalues and eigenvectors of a couple of matrices which commute. This is a typical problem in Quantum Mechanics: when you have two observables represented by matrices, A and B, if they both commute, i.e. if AB=BA, then you can find a common basis associated to them.

Is there any option in Mathematica that can do this?

Thanks in advance!!!

Rafa.

2

Might try getting eigensystem of the product. If the eigenvalues are distinct then you know those eigenvectors are simultaneous for both matrices.

The eigenvalues are not common to the matrices by the way, just the eigenvectors. Mentioning that because the language in the post was on the ambiguous side in regard to that detail.