Weinstein–Aronszajn identity

{{Redirect-distinguish|Sylvester's determinant theorem|Sylvester's determinant identity}}{{Short description|For two suitable matrices, A and B, I+AB and I+BA have the same determinant}}

In mathematics, the Weinstein–Aronszajn identity states that if $A$ and $B$ are matrices of size {{math|m × n}} and {{math|n × m}} respectively (either or both of which may be infinite) then,

provided $AB$ (and hence, also $BA$ ) is of trace class,

: $\det(I_m + AB) = \det(I_n + BA),$

where $I_k$ is the {{math|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.{{citation|title=An Introduction to Grids, Graphs, and Networks|first=C.|last=Pozrikidis|publisher=Oxford University Press|year=2014|isbn=9780199996735|page=271|url=https://books.google.com/books?id=Ws_RAgAAQBAJ&pg=PA271}}

Let $M$ be a matrix consisting of the four blocks $I_m$ , $A$ , $B$ and $I_n$ :

: $M = \begin{pmatrix} I_m & A \\ B & I_n \end{pmatrix}.$

Because {{math|I_m}} is invertible, the formula for the determinant of a block matrix gives

: $\det\!\begin{pmatrix} I_m & A \\ B & I_n \end{pmatrix} = \det(I_m) \det(I_n - B I_m^{-1} A) = \det(I_n - BA).$

Because {{math|I_n}} is invertible, the formula for the determinant of a block matrix gives

: $\det\!\begin{pmatrix} I_m & A\\ B & I_n \end{pmatrix} = \det(I_n) \det(I_m - A I_n^{-1} B) = \det(I_m - AB).$

Thus

: $\det(I_n - B A) = \det(I_m - A B).$

Substituting $-A$ for $A$ then gives the Weinstein–Aronszajn identity.

Applications

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

: $\det(AB - \lambda I_m) = (-\lambda)^{m - n} \det(BA - \lambda I_n).$

It follows that the non-zero eigenvalues of $AB$ and $BA$ 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.{{cite web|url=http://terrytao.wordpress.com/2010/12/17/the-mesoscopic-structure-of-gue-eigenvalues/ |title=The mesoscopic structure of GUE eigenvalues | What's new |website=Terrytao.wordpress.com |date= 18 December 2010|accessdate=2016-01-16}}

References

Category:Determinants

Category:Matrix theory

Category:Theorems in linear algebra