TITLE: Determinants (and traces) of linear maps of matrices
QUESTION [8 upvotes]: Let $k$ be a field or a commutative ring with unit and let $F:M_n(k)\to M_n(k)$ be a $k$-linear map. Suppose that $F$ is given in the form $F(X) = A_1XB_1 + \cdots + A_m X B_m$ for some $A_i,B_i\in M_n(k)$ (note that any $k$-linear $F$ can be written in this form, though not uniquely). 

Is there a formula that allows one to determine $\mathrm{det}(F)$ (no pun intended) or $\mathrm{tr}(F)$ directly from the matrices $A_i, B_i$ ($1\leq i\leq m$), i.e. without having to compute a representation matrix of $F$?

REPLY [5 votes]: We have $F(X)=\sum_i A_i X B_i = \sum_i (B_i^T \otimes A_i) vec(X)$ (see here), i.e. $F \sim \sum_i (B_i^T \otimes A_i)$. Because of some formulas here we have $tr(F)=\sum_i tr(A_i)tr(B_i)$. For the determinant I don't think there is a nice formula.