Lets denote the Kronecker product by $\otimes$ and the vectorization of a matrix $Y$ by $\mathrm{vec}(Y)$.
Given $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$, where $n\geq m$. What is
$$\mathrm{trace}(BB^TY^{-1}),$$where $\mathrm{vec}(Y)=(A\otimes A-I)^{-1}\mathrm{vec}(BB^T)$? (Assume all eigenvalues of $A$ are larger than $1$)
Currently, given specific $A$ and $B$, I have to compute the $\mathrm{vec}(Y)$, then retrieve matrix $Y$, then take the inverse of $Y$ and then compute the $\mathrm{trace}(BB^TY^{-1})$. However, since $Y$ depends on $A$ and $B$, I feel that there must be a way to write down the $\mathrm{trace}(BB^TY^{-1})$ in terms of $A$ and $B$ only. I tried to rewrite it like this:
$$\mathrm{trace}(BB^TY^{-1})=\mathrm{vec}(BB^T)^T\mathrm{vec}(Y^{-1}),$$
but I couldn't find what is the vectorization of the inverse matrix, i.e., $\mathrm{vec}(Y^{-1})$?