# Weinstein–Aronszajn identity

In mathematics, the Weinstein–Aronszajn identity states that if $A$ and $B$ are matrices of size m × n and n × m respectively (either or both of which may be infinite) then, provided $AB$ (and hence, also $BA$ ) is of trace class,

$\det(I_{m}+AB)=\det(I_{n}+BA),$ where $I_{k}$ is the k × k identity matrix.

It is closely related to the matrix determinant lemma and its generalization. It is the determinant analogue of the Woodbury matrix identity for matrix inverses.

## Proof

The identity may be proved as follows. Let $M$  be a matrix comprising the four blocks $I_{m}$ , $-A$ , $B$  and $I_{n}$ .

$M={\begin{pmatrix}I_{m}&-A\\B&I_{n}\end{pmatrix}}.$

Because Im is invertible, the formula for the determinant of a block matrix gives

$\det {\begin{pmatrix}I_{m}&-A\\B&I_{n}\end{pmatrix}}=\det(I_{m})\det \left(I_{n}-BI_{m}^{-1}(-A)\right)=\det(I_{n}+BA).$

Because In is invertible, the formula for the determinant of a block matrix gives

$\det {\begin{pmatrix}I_{m}&-A\\B&I_{n}\end{pmatrix}}=\det(I_{n})\det \left(I_{m}-(-A)I_{n}^{-1}B\right)=\det(I_{m}+AB).$

Thus

$\det(I_{n}+BA)=\det(I_{m}+AB).$

## Applications

Let $\lambda \in \mathbb {R} \setminus \{0\}$ . The identity can be used to show the somewhat more general statement that

$\det(AB-\lambda I_{m})=(-\lambda )^{m-n}\det(BA-\lambda I_{n}).$

It follows that the non-zero eigenvalues of $AB$  and $BA$  are the same.

This identity is useful in developing a Bayes estimator for multivariate Gaussian distributions.

The identity also finds applications in random matrix theory by relating determinants of large matrices to determinants of smaller ones.