Lower bound on det(A'DA) with A tall and D diagonal?

Wrukproek

Hi all,

I searched all other threads but was unable to find something useful. Here is my question: I am looking for a lower bound on

det(A'DA),

where A is a MxN with M>=N (possibly tall and skiny) and D is diagonal, non-negative and of dimension M. In particular, I am looking for ways that separate the properties of the matrices A and D in this expression. I know that for M=N, we can rewrite it as

(1) det(A'*D*A) = det(A)^2*det(D),

which is very helpful. However, for the more general case with M>N, which is the one I am interested in, I am unable to derive a useful equality or lower bound (which would also be fine) that separates the matrices A and D in (1). Upper bounds are not useful in my particular application.

Thanks a lot for your help,
Wrukproek