Weinstein–Aronszajn identity

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

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

where ${\displaystyle I_{k}}$ 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.[1] Let ${\displaystyle M}$ be a matrix comprising the four blocks ${\displaystyle I_{m}}$, ${\displaystyle -A}$, ${\displaystyle B}$ and ${\displaystyle I_{n}}$.

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

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

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

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

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

Thus

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

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.[2]

References

1. Pozrikidis, C. (2014), An Introduction to Grids, Graphs, and Networks, Oxford University Press, p. 271, ISBN 9780199996735
2. "The mesoscopic structure of GUE eigenvalues | What's new". Terrytao.wordpress.com. Retrieved 2016-01-16.