McCoy's Theorem

If two Square $n\times n$ Matrices A and B are simultaneously upper triangularizable by similarity transforms, then there is an ordering $a_1$, ..., $a_n$ of the Eigenvalues of A and $b_1$, ..., $b_n$ of the Eigenvalues of B so that, given any Polynomial $p(x,y)$ in noncommuting variables, the Eigenvalues of $p(A,B)$ are the numbers $p(a_i,b_i)$ with $i=1$, ..., $n$. McCoy's theorem states the converse: If every Polynomial exhibits the correct Eigenvalues in a consistent ordering, then A and B are simultaneously triangularizable.

References

Luchins, E. H. and McLoughlin, M. A. ``In Memoriam: Olga Taussky-Todd.'' Not. Amer. Math. Soc. 43, 838-847, 1996.