TITLE: Is there a name referring to this result?
QUESTION [8 upvotes]: 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?

REPLY [8 votes]: It does: it is a special case of Sylvester's determinant theorem.