Let's say I have two covariance matrices $A$ and $B$ (so they're both positive semi-definite), What kind of transformations can I apply on either one of them or both without loosing the positive-definite aspect in the resuling matrix ?
Mainly, I want to do this :
$f(A)=C$ or $g(A,B)=D$
and still have $C$ and $D$ positive semi-definite.
My ultimate goal is to use these transformations as a mutation operator $(f(A)=C)$ or a crossover operator $(g(A,B)=D)$ in a genetic algorithm. So keeping the positive semi-definite property is important to avoid making lots of crossovers and mutations without ending-up with suitable results since I want the end result to be a covariance matrix.
if $B$ is non singular square matrix you can do this:
$g(A,B)=B^{-1}AB$ the result of this function is positive semi-definite if $A$ was.
if $A$ and $B$ are positive semi-definite than you can build a block matrix $D = \left[\begin{matrix}A&0\\0&B \end{matrix}\right]$ which is also positive semi-definite