Is there a name referring to this result?

Question

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?

@MTurgeon: My question came from preliminary differential geometry, when considering the Riemannian volume form $\sqrt{g}$ of the metric on a graph of a multivariable function. — Sangchul Lee, Apr 18 '12 at 01:53
I thought at first that there was no reason for it to have a name. But then I noticed it was familiar, thus my answer below. So I was wrong. — M Turgeon, Apr 18 '12 at 01:57
See this question for proofs of Sylvester's determinant identity. — Bill Dubuque, Apr 18 '12 at 02:40

score 8 · Accepted Answer · edited Apr 18 '12 at 01:56

8

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

edited Apr 18 '12 at 01:56

J. M. ain't a mathematician

75,051

answered Apr 18 '12 at 01:53

M Turgeon

10,419

1

Surely this is what I've been looking for! Thanks so much! – Sangchul Lee Apr 18 '12 at 01:57

Is there a name referring to this result?

1 Answers1