In mathematics, the Weinstein–Aronszajn identity states that if
are matrices of size m × n and n × m respectively (either or both of which may be infinite) then, provided
It is closely related to the matrix determinant lemma and its generalization.
be a matrix consisting of the four blocks
: 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
The identity can be used to show the somewhat more general statement that It follows that the non-zero eigenvalues of
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.
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