How does composition affect eigendecomposition? What relationship is there between the eigenvalues and vectors of linear operator $T$ and the composition $A T$ or $T A$? I'm also interested in analogous results for SVD.

Friedland has proved the following over the complex field:
If the principal minors of $A$ are not zero, then for every set of $n$ numbers $\lambda_1,\dots,\lambda_n$ there exist a diagonal matrix $B$ such that $BA$ has $\lambda_i$'s as eigenvalues. 
Later Dias da Silva extended it to any arbitrary algebraically closed field.
