Let $S$ and $T$ be two linear transformations on $\mathbb{R}^n$. We can find the eigenvalues and eigenvectors of $S$ and $T$. I am trying to find out the relation(s) among the eigenvalues and eigenvectors of $S$, $T$ and $S\circ T$. Is there any relation of this kind?



What are the eigenvalues and eigenvectors of a composition of two arbitrary linear transformations?
This is a very difficult problem, the relations are not equalities but inequalities,
and in an important special case it was solvedd in


S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum
Schubert calculus, math.AG/9712013, Math. Res. Lett. 5 (1998), 817–836. MR 2000a:14066


