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 Web Resource. https://mathworld.wolfram.com/McCoysTheorem.html