Common Eigenvalues of a Commuted Matrix Product

Question

Let $A \in \mathbb{R}^{n \times m}$, $B \in \mathbb{R}^{m \times n}$. Can we prove that $AB$ and $BA$ share all nonzero eigenvalues?

Based on comments here, this appears to be possible.

See, e.g., here. The question considers square $A$ and $B$ but some proofs work also for rectangular matrices. — Algebraic Pavel, Jun 12 '15 at 15:51

score 2 · Accepted Answer · answered Jun 12 '15 at 15:55

2

Hint: if $AB u = \lambda u$, what can you say about $BA (Bu)$?

answered Jun 12 '15 at 15:55

Robert Israel

1 Answers1