find the eigenvalues of a linear operator T Let $A$ be $m*m$ and B be $n*n$ complex matrices, and consider the linear operator $T$ on the space $C^{m*n}$ of all  $m*n$ complex matrices defined by $T(M) = AMB$.
-Show how to construct an eigenvector for $T$ out of a pair of column vectors $X, Y$, where $X$ is an eigenvector for $A$ and $Y$ is an eigenvector for $B^t$.
-Determine the eigenvalues of $T$ in terms of those of $A$ and $B$
-Determine the trace of this operator

There is a standard way to do this kind of exercise.  Firstly assume that $ A$ and $ B$ are diagonal.  Then a short calculation shows that the eigenvalues are as given in previous solutions, i.e., the pairwise products of those of these matrices.  The result then holds for diagonalisable matrices by a suitable choice of bases.  The easiest way to get the final version is to use the fact that the diagonalisable matrices are dense in all matrices and employ a continuity argument involving the characteristic polynomials.
