8
$\begingroup$

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?

  • 0
    @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.2012-04-18
  • 0
    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.2012-04-18
  • 2
    See [this question](http://math.stackexchange.com/questions/17831/sylvesters-determinant-identity) for proofs of Sylvester's determinant identity.2012-04-18

1 Answers 1

8

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

  • 1
    Surely this is what I've been looking for! Thanks so much!2012-04-18