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Weinstein–Aronszajn identity

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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.

Proof

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The identity may be proved as follows.[1] Let be a matrix consisting of 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

Substituting for then gives the Weinstein–Aronszajn identity.

Applications

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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]

References

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  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.