Let $A\in \mathbb R^{m\times n}$ be arbitrary. Suppose $W\in \mathbb R^{m\times m}$ and $Y\in \mathbb R^{n\times n}$ are orthogonal.
- Suppose $W$ and $Y$ are not orthogonal but simply nonsingular. Do the singular values of $A$ and $WAY$ still coincide? How about their ranks?
Note the first part of the question was answered here (with $W=P$ and $Y=Q$). Thus, not included it. Regarding the question above:
One would think to still compare the matrices $AA^T$ and $(WAY)(WAY)^T = WAYY^TA^TW^T$. It seems these matrices are not similar and thus do not have the same singular values nor same rank. This is becuase $AYY^TA^T$ is not the same as $AYY^TA^T = AA^T$ when $Y$ is orthogonal (same post).
Is the logic sound? Help is great!
The rank yes, because in fact two matrices $A,B\in\Bbb F^{n\times m}$ have the same rank if and only if there are $X\in GL(n,\Bbb F)$ and $Y\in GL(m,\Bbb F)$ such that $B=XAY$.
Singular values obviously no, see for instance the case where $W=3I$ and $Y=5I$.