If I have a diagonal matrix $D$ and a positive-definite symmetric matrix $C$, is there a formula for the square root of the product, $(DC)^{1/2}$? Also, $DC \neq CD$.
What I have so far is \begin{align} (DC)^{1/2} &= D^{1/2}(D^{1/2} C D^{1/2})^{1/2}D^{-1/2} \end{align} Can the middle term be broken down any further? I am trying to use $C^{1/2}$ or $C^{-1/2}$ somehow but am unable to.
Edit: I am trying to see if $D^{1/2}CD^{1/2}$ can be written as the sum of a diagonal matrix with an outer product, $(D+cc^T)^{1/2}$, if that would make things simpler. If the diagonal is the same i.e. the identity matrix, I could calculate $(I+cc^T)^{1/2}$ but having different values in the diagonal makes it more difficult.
I could calculate $(D+cc^T)^{1/2}$ with a Cholesky rank-one update, but I'm looking for a non-iterative formula.
In order to find this, I would need to find an idempotent matrix of the form $(D-A)$ where $A^2=A$. $(I-A)$ is idempotent but I think it is impossible when it is not the identity matrix.