If two square 
matrices 
and 
are simultaneously upper triangularizable by similarity transforms,
then there is an ordering 
, ..., 
of the eigenvalues of 
and 
, ..., 
of the eigenvalues of 
so that, given any polynomial 
in noncommuting variables, the
eigenvalues of 
are the numbers 
with 
, ..., 
. McCoy's theorem states the converse: If every polynomial
exhibits the correct eigenvalues in a consistent ordering,
then 
and 
are simultaneously triangularizable.
McCoy's Theorem
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References
Luchins, E. H. and McLoughlin, M. A. "In Memoriam: Olga Taussky-Todd." Not. Amer. Math. Soc. 43, 838-847, 1996.Referenced on Wolfram|Alpha
McCoy's TheoremCite this as:
Weisstein, Eric W. "McCoy's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/McCoysTheorem.html