Weinstein–Aronszajn identity

In mathematics, the Weinstein–Aronszajn identity states that if and are matrices of size m × n and n × m respectively (either or both of which may be infinite) then, provided (and hence, also ) is of trace class,

where 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.

ProofEdit

The identity may be proved as follows.[1] Let   be a matrix comprising the four blocks  ,  ,   and  .

 

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

 

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

 

Thus

 

ApplicationsEdit

Let  . The identity can be used to show the somewhat more general statement that

 

It follows that the non-zero eigenvalues of   and   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]

ReferencesEdit

  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.