Least nonzero singular value of $A^{-1} P$ for some invertible matrix $A$ corresponds to the reciprocal of the largest singular value of $A \circ P$?

Question

Let $A$ be $n \times n$ real invertible matrix, and $P : \mathbb{R}^n \to \mathbb{R}^n$ be some orthogonal projection.

Then, what would be the relation between the smallest nonzero singular value of $A^{-1} \circ P$ and the largest singular value of $A \circ P$?

I suspect that they are related by reciprocal...but I cannot really justify my guess.. Could anyone please help me?

Here $\circ$ just means the matrix product. I have been considering them as linear mappings..

Answer 1

They aren’t reciprocal to each other in general. In fact, for any $n\ge2$ and $c>0$, if we put $$ A=\pmatrix{0&c^{-1}\\ 1&0\\ &&I_{n-2}} \quad\text{and}\quad P=\pmatrix{1&0\\ 0&0\\ &&\mathbf 0_{(n-2)\times(n-2)}}, $$ the product of the largest singular value of $AP$ and the smallest nonzero singular value of $A^{-1}P$ will be equal to $c$.