Let $X$ be an $n\times p$ real matrix with column rank $k$, where $0<k<p<n$, and let $A$ be a semi-orthogonal matrix (the columns are orthonormal) such that $A'X=0$, i.e. the column space of $A$ is the orthogonal complement to the column space of $X$. Let $C$ be an arbitrary positive definite $n\times n$ matrix.
Question: Can the determinant $\vert A'CA \vert $ be expressed as a function of $X$ and $C$?
When $p=k$ so that $X$ is full column rank, the answer is yes and one can write $\vert A'CA\vert = \vert A'A\vert\vert C\vert\vert X'C^{-1}X\vert/\vert X'X\vert$. This is equation (9) in LaMotte. I have thought about relating the non-zero singular values of $X$ and $C$ to those of $A'CA$ but haven't succeeded.
This question is related to a question of more statistical nature posted on stats.stackexchange in the sense than an answer to the present question may be a help on the way to finding an answer to the statistical question.
