Hey kkar.
I'm just wondering if you can use the change of basis formula and then diagonalize the resulting matrix?
Consider two Hermitian matrices A1 and A2 that commute (in mathematica syntax, every set is a row {}): A1={{1,0,1},{0,0,0},{1,0,1}} and A2={{2,1,1},{1,0,-1},{1,-1,2}}. The matrix A1 had eigenvalues and orthonormal eigenvectors: lambda1=2, lambda2=0=lambda3 corresponding to v1=1/sqrt(2){{1},{0},{1}}, v2=1/sqrt(2){{-1},{0},{1}}, v3=1/sqrt(2){{0},{1},{0}}. In the basis of these eigenvectors find A2 and then your result to find the eigenvalues of A2.
I did the first part by creating let's call it matrix V, a matrix of the eigenvectors as the columns and then doing the following: P^-1A2P. That gives me the matrix, let's call it B2={{3,0,0},{0,1,-sqrt(2)},{0,-sqrt(2),0}}
But how do I do the second part?
I also toyed (am toying) with that idea that intuitively seems very promising. The thing is, how do I go about diagonalizing the resulting matrix if I don't have it's eigenvectors? Is it something about the hermitivity of the two matrices that might allow me to use a version of A1's eigenvector?