Modifying a matrix equation

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C.Koca

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C.Koca Asks: Modifying a matrix equation
I have a matrix equation of the form

$$ Q=SBS+SBA+HSBZ+ABS+ABA+ABZ+ZBS+ZBA+ZBZ. $$

If it weren't for the $H$ matrix in the third term, the whole thing would be

$$ Q^\prime=(S+A+Z)B(S+A+Z). $$

As it is, I was able to reduce it to

$$ Q=(S+A+Z)B(S+A+Z)+(H-I)SBZ, $$ but I want to eliminate the addition, if possible. In other words, I am looking for an expression like

$$ Q=f(S,A,H,Z)Bg(S,A,H,Z), $$ where $f(.)$ and $g(.)$ are matrix functions.

All matrices are square and real. $S$ is symmetric and $H$ is a diagonal, i.e., they commute.

I am looking for some $f(.)$ and $g(.)$. I will be calculating $$ Q^{(n)}=f^n(S,A,H,Z) B g^n(S,A,H,Z), $$ and I will be using SVD to proceed. Thus eliminating the addition helps me a lot.

Thanks!

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