Condition for product of tow rectangular matirx is diagonalizable?

Question

Let $A$ and $b$ be $m \times n$ matrices, it seems the product $A'B$ is diagonalizable only if $A$ and $B$ share the same left and right singularvectors. Is it true ?. How can I prove it, in case it is true ?

Answer 1

First, it does not make sense to talk about eigenvalues unless $m=n$.

Second, consider the following example $$ A=\pmatrix{2&1\\2&-1}, B = A\pmatrix{0&1\\1&0} = \pmatrix{1&2\\-1&2}, $$ then $$ A^TB = \pmatrix{4&0\\0&-4},$$ which is clearly diagonalizable. However, $A$ and $B$ do not share any left or right eigenvector, as $\pmatrix{1\\1}$ is no eigenvector of these matrices.