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

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

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

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

Thus

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

Applications[]

Let ${\displaystyle \lambda \in \mathbb {R} \setminus \{0\}}$. The identity can be used to show the somewhat more general statement that

${\displaystyle \det(AB-\lambda I_{m})=(-\lambda )^{m-n}\det(BA-\lambda I_{n}).}$

It follows that the non-zero eigenvalues of ${\displaystyle AB}$ and ${\displaystyle BA}$ 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.^[2]

References[]

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