Weinstein-Aronszajn identity

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the Weinstein-Aronszajn identity

1. (mathematics) An identity stating 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.