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, 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 consisting of the four blocks I_m, A, B and I_n: Because Im is invertible, the formula for the determinant of a block matrix gives Because In is invertible, the formula for the determinant of a block matrix gives Thus Substituting -A for A then gives the Weinstein–Aronszajn identity.

Applications

Let. The identity can be used to show the somewhat more general statement that 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.