My question is actually general: how can one "factorize" a matrix? In particular, given an arbitrary matrix $M$, can one write $$M=ABC$$ for some non-zero matrices $A, B, C$? If yes, is there some way of finding these matrices ("factors")?
EDIT: After reading some nice comments, I present here the conditions on matrices A, B and C, and these are: $A = \sqrt{a}$, $C=\sqrt{c}$ and $B= u v^\dagger$ where $u$ and $v$ are some column vectors and $*$ denotes the combined operation of complex-conjugation and transposition.