Let $\Delta$ be a diagonal, non-invertible matrix with complex entries. Is it possible to come up with a matrix $M$ such that
$U\equiv M.\Delta$
is unitary?
Though I don't know about the proof, I heard that there is a theorem stating that any invertible matrix $A$ with complex entries can be written as
$A=U.T$
where $T$ is upper triangular. Since diagonal matrices are a subset of upper triangular, I basically want to know if the converse of this theorem above is true, and if there's an algorithmic way to find $A$ such that $A.T^{-1}$ defines a unitary.
PS.: $M$ has the same dimension as $\Delta$