The low-rank eigenvalue problem

Abstract

The nonzero eigenvalues of $AB$ are equal to those of $BA$: an identity that holds as long as the products are square, even when $A,B$ are rectangular. This fact naturally suggests an efficient algorithm for computing eigenvalues and eigenvectors of a low-rank matrix $X= AB$ with $A,B^T\in\mathbb{C}^{N\times r}, N\gg r$: form the small $r\times r$ matrix $BA$ and find its eigenvalues and eigenvectors. For nonzero eigenvalues, the eigenvectors are related by $ ABv = \lambda v \Leftrightarrow BAw = \lambda w $ with $w=Bv$, and the same holds for Jordan vectors. For zero eigenvalues, the Jordan blocks can change sizes between $AB$ and $BA$, and we characterize this behavior.