MATRIX VIETA THEOREM

Dmitry Fuchs and Albert Schwarz

Abstract. We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-k matrices.

Specifically, we prove that if X_{1}, …, X_{n} are solutions of an algebraic matrix equation X^{n} +A_{1}X^{n−1} +⋯+A_{n} = 0, independent in the sense that they determine the coefficients A_{1}, …, A_{n}, then the trace of A_{1} is the sum of the traces of the X_{i}, and the determinant of A_{n} is, up to a sign, the product of the determinants of the X_{i}. We generalize this to arbitrary rings with appropriate structures.

This result is related to and motivated by some constructions in non-commutative geometry.

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