Weinstein–Aronszajn identity

In mathematics, the Weinstein–Aronszajn identity states that if and are matrices of size and respectively then,
provided is of trace class,
where is the 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 be a matrix consisting of the four blocks,, and :
Because is invertible, the formula for the determinant of a block matrix gives
Because is invertible, the formula for the determinant of a block matrix gives
Thus
Substituting for 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 and 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.