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McCoy's Theorem


If two square n×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.


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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.

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McCoy's Theorem

Cite this as:

Weisstein, Eric W. "McCoy's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/McCoysTheorem.html

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