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For any real $m \times n$ matrix $A$, it seems that $\det(I_n + A^{T}A) = \det(I_m + AA^{T}) $ always holds, where $I_n$ is the identity matrix of size $n$.

Though I have not tried to prove this yet, I'm sure it is a part of well-known results in linear algebra. So my question is, what is the name referring to this fact, and where can I find a reference to it?

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    See [this question](http://math.stackexchange.com/questions/17831/sylvesters-determinant-identity) for proofs of Sylvester's determinant identity.2012-04-18

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It does: it is a special case of Sylvester's determinant theorem.

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    Surely this is what I've been looking for! Thanks so much!2012-04-18