The Weinstein-Aronszajn identity is a result in linear algebra stating that for two matrices A and B of compatible sizes m by n and n by m, where either or both may be infinite, provided the product AB is of trace class, the matrices I plus AB and I plus BA have the same determinant. It is closely related to the matrix determinant lemma and its generalizations, and it serves as the determinant analogue of the Woodbury matrix identity for matrix inverses. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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1. Weinstein-Aronszajn identity (Wikipedia)
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In mathematics, the Weinstein-Aronszajn identity states that if A {\displaystyle A} and B {\displaystyle B} are matrices of size m × n and n × m respectively (either or both of which may be infinite) then, provided A B {\displaystyle AB} (and hence, also B A {\displaystyle BA} ) is of trace class, det ( I m + A B ) = det ( I n + B A ) , {\displaystyle \det(I_{m}+AB)=\det(I_{n}+BA),} where I k {\displaystyle I_{k}} is the k × k identity matrix.
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