Let $A$ be a fixed positive semi-definite symmetric $m\times m$ matrix, and let $p$ be a fixed positive integer. Let $Z$ vary over all $m\times p$ matrices with orthonormal columns, and denote the transpose of $Z$ by $Z^T$. How does one prove that the minimum of $\mbox{det}(Z^T.A.Z)$ is the product of the $p$ smallest eigenvalues of $A$?

I've seen this quoted in a few places, but the references cited have turned out to be either vacuous ("a little algebra shows") or encrusted with so much generality that I can't see through to the core of the proof. Does anyone have a *good* reference or a proof that is short enough to be given in this forum? Not surprisingly, I can prove it for $p=1$ and for $p=m$.

And is there an analogous result when "minimum" is replaced by "maximum"?

Does anyone know to whom these results are due? They are probably pretty ancient.