If I have a matrix M which can be decomposed as:
$M = DH$
where $D$ is a diagonal matrix and $H$ is another matrix with known positive semi-definite square root $H^{1/2}$, does this give a formula or method for finding $M^{1/2}$?
Some additional details for the specific problem that I'm looking at, both D and H are positive semi-definite, Hermitian and have trace 1. In general they are non-commuting.
Best Answer
By "the unique non-negative square root", I presume that you mean a positive semidefinite square root. But if this is the case, your question is not well posed: if $M^{1/2}$ is positive semidefinite, so is $M$. Hence $M$ is Hermitian. Yet if $M=DH$ where $M$ is Hermitian, $D$ is diagonal and $H$ is Hermitian, then $D$ and $H$ must commute.
Edit: At any rate, I don't think there is any easy formula for calculating a square root of $DH$ in general. When $D$ is nonsingular, we can calculate $(DH)^{1/2}$ as $$ (DH)^{1/2} = D^{1/2} (D^{1/2} H D^{1/2})^{1/2} D^{-1/2},, $$ where $(D^{1/2} H D^{1/2})^{1/2}$ is the unique positive semidefinite square root of $D^{1/2} H D^{1/2}$. This $(DH)^{1/2}$ is the unique square root of $DH$ with nonnegative eigenvalues. But to calculate it, you need to find a square root of $D^{1/2} H D^{1/2}$, and to do so, knowing $H^{1/2}$ is not really useful.