Matrix Vieta Theorem
Open Access Publications from the University of California

## Matrix Vieta Theorem

• Author(s): Fuchs, Dmitry
• Schwarz, Albert
• et al.

## Published Web Location

https://arxiv.org/pdf/math/9410207.pdf
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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,\dots ,X_n\$ are solutions of an algebraic matrix equation \$X^n+A_1X^{n-1}+\dots +A_n=0,\$ independent in the sense that they determine the coefficients \$A_1,\dots ,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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