If you have two matrices A and B such as through singular value decomposition:
AB = PSV
And any invertible matrix T such as
AB = $PS^{1/2}TT^{-1}S^{1/2}V$
Could you prove that B = $T^{-1}S^{1/2}V$
This question might be too out of context, so to situate it here is the page where I encontered the problem, it is from the book Biological Learning and Control by Sandro Mussa Ivaldi:
The rank of $AB$ is less than or equal to the minimum of the rank of $A$ and $B$ (this is known as Sylvester's theorem I think). And the rank of $S^{1/2}$ is the same as the rank of $AB$ (SVD theory: rank of $AB$ equals the number of nonzero singular values). The equation $B=T^{-1}S^{1/2}V$ basically states that $B$ and $S^{1/2}$ are equivalent (matrix equivalence) and hence must have the same rank. But since the rank of $AB$ may very well be less than the rank of $B$ we see that this cannot be true in general. A trivial counter example: let $A=0$ and $B=I$.