Weinstein–Aronszajn identity
For two suitable matrices, A and B, I+AB and I+BA have the same determinate / From Wikipedia, the free encyclopedia
"Sylvester's determinant theorem" redirects here. Not to be confused with Sylvester's determinant 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.