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 comprising 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

Applications

This identify 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.