Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric matrices. Suppose that $AB$ is non-singular. Let us define


$$F := (AB)^{-1} CB (AB)^{-1}P(AB)^{-\top} + (AB)^{-1}P(AB)^{-\top}(CB)^\top (AB)^{-\top}.$$


Next, consider the scalar function
$$
c(X)=\mathrm{tr}(B^{\top}XBF)
$$
where $X\in\mathcal{S}^n$ is such that $B^{\top}XB\ne 0$ and $\mathrm{tr}(\cdot)$ denotes the trace operator.



> 
> Is it possible to find a suitable $X$ such that $c(X)$ turns into a quadratic expression, i.e., something of the form
>  $$
> c(X)=\mathrm{tr}(L^\top Y L),
> $$
>  where $L$ and $Y$ are functions of $A,B,C,P$ and $Y$ is positive definite?
> 
> 
> 


*Note.* I made several attempts with different $X$'s, but without any success yet. Hence, I wonder whether there is a rationale to follow either to get a positive answer to my question or to prove that this is actually not doable.



Completing the square of a matrix expression
Nope. That would mean $c(X)$ is always nonnegative, but that can't be that way (unless it's always $0$) because it switches sign if you replace $C$ with $-C$.


