View source for 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 determinate}}
In [[mathematics]], the '''Weinstein–Aronszajn identity'''  states that if <math>A</math> and <math>B</math> are [[Matrix (mathematics)|matrices]] of size {{math|''m'' × ''n''}} and {{math|''n'' × ''m''}} respectively (either or both of which may be infinite) then,
provided <math>AB</math> (and hence, also <math>BA</math>) is of [[trace class]],

:<math>\det(I_m + AB) = \det(I_n + BA),</math>

where <math> I_k </math> 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 inverse]]s.

==Proof==
The identity may be proved as follows.<ref>{{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}}</ref> 
Let <math> M</math> be a matrix consisting of the four [[Block matrix|blocks]] <math>I_m</math>, <math>A</math>, <math>B</math> and <math>I_n</math>:

:<math>M = \begin{pmatrix} I_m & A \\ B & I_n \end{pmatrix}. </math>

Because {{math|''I''<sub>''m''</sub>}} is [[invertible matrix|invertible]], the formula for the [[Determinant#Block matrices|determinant of a block matrix]] gives

:<math>\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). </math>

Because {{math|''I''<sub>''n''</sub>}} is invertible, the formula for the determinant of a block matrix gives

:<math>\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).</math>

Thus
:<math>\det(I_n - B A) = \det(I_m - A B).</math>
Substituting <math>-A</math> for <math>A</math> then gives the Weinstein–Aronszajn identity.

==Applications==
Let <math>\lambda \in \mathbb{R} \setminus \{0\}</math>. The identity can be used to show the somewhat more general statement that

: <math>\det(AB - \lambda I_m) = (-\lambda)^{m - n} \det(BA - \lambda I_n).</math>

It follows that the non-zero [[eigenvalue]]s of <math>AB</math> and <math>BA</math> are the same.

This identity is useful in developing a [[Bayes estimator]] for [[multivariate Gaussian distribution]]s.

The identity also finds applications in [[random matrix theory]] by relating determinants of large matrices to determinants of smaller ones.<ref>{{cite web|url=http://terrytao.wordpress.com/2010/12/17/the-mesoscopic-structure-of-gue-eigenvalues/ |title=The mesoscopic structure of GUE eigenvalues &#124; What's new |website=Terrytao.wordpress.com |date= 18 December 2010|accessdate=2016-01-16}}</ref>

==References==
{{reflist}}

{{DEFAULTSORT:Weinstein-Aronszajn identity}}
[[Category:Determinants]]
[[Category:Matrix theory]]
[[Category:Theorems in linear algebra]]


{{linear-algebra-stub}}