# Two commuted Matrices with degenerate eigenvalues

I have two huge commuted Matrices A and B, each one of them contains a large amount of degenerate eigenvalues. Now I compute the eigenvalues and eigenvectors of A. Since [A,B] = 0, if eigenvalue of A is not degenerate, the eigenvector of A should also be the eigenvector of B. However if some eigenvalues of A are degenerate, after I compute these eigenstates using program numerically, these eigenstates might not be the eigenstates of B.

Now I only want those eigenvectors with one specific eigenvalue(say b, which is a number) of B, then how can I get the eigenstates of B with eigenvalue b in these degenerate eigenstates of A? My purpose is trying to reduced matrix A into the subspace of those states with eigenvalue of B, b.

• I think you mean "commuting matrices". Do you mean that $AB = BA$? – Omnomnomnom May 10 '17 at 14:26
• For those who don't know: "degenerate" = "repeated" – Omnomnomnom May 10 '17 at 14:28
• I don't understand the actual question at all. I don't understand " how can I get the eigenstates of B with eigenvalue b in these degenerate eigenstates of A" – Omnomnomnom May 10 '17 at 14:48
• Sorry for the confusion. Since these repeated eigenstates of A might not be the eigenstates of B (when you do it numerically), then I want to find out what is the eigenstates of B in this subspace spanned by these repeated eigenstates of A. In particular, I want those eigenstates with eigenvalue b of B. – JoeJackJessieJames May 10 '17 at 23:44
• So given a specific eigenvalue $b$ of $B$, you're looking for eigenstates for $B$ (associated with $B$) that are also eigenstates of $A$. Is that correct? – Omnomnomnom May 11 '17 at 0:59

Given eigenvalues $a$ of $A$ and $b$ of $B$, you're looking for a vector $x$ such that $Ax = ax$ and $Bx = bx$. That is, you're looking for a vector $x$ such that $(A - aI)x = (B - bI)x = 0$ (where $I$ is the identity matrix). In other words, you're looking for a vector $x$ such that $$\pmatrix{A - a I\\ B - bI}x = 0$$ So, it is sufficient to find a basis of the nullspace of the block-matrix $\pmatrix{A - a I\\ B - bI}$.