Weinstein–Aronszajn identity

Weinstein–Aronszajn identity   (Redirected from Sylvester's determinant theorem) Jump to navigation Jump to search "Sylvester's determinant theorem" redirects here. Not to be confused with Sylvester's determinant 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} (and hence, also {displaystyle BA} ) is of trace class, {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 Im 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 In 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}Bright)=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 ^ Pozrikidis, C. (2014), An Introduction to Grids, Graphs, and Networks, Oxford University Press, p. 271, ISBN 9780199996735 ^ "The mesoscopic structure of GUE eigenvalues | What's new". Terrytao.wordpress.com. Retrieved 2016-01-16.

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