B,C are two tall matrices but are not assumed to have full column rank. Show that $BB^H=CC^H$ if and only if $C=BQ$ for some unitary matrix Q with k x k dimension.
I try to use EVD on both $CC^H$ and $BB^H$. it seems like that Q is a matrix whose columns are the eigenvectors of $CC^H$. However, I cannot find the exact Q matrix.
some thoughts： if B, C is assumed to have full column rank, then $CC^H$ is invertible. Then I can find $Q = C(C^{H}C)^{-1}$ so that C=BQ and Q can be confirmed to be unitary. but it does not follow the requirements of the question.
Any hints or comments? thanks