Celebratio Mathematica — Fuchs — Complete Bibliography

Complete Bibliography

Works connected to Evgeniĭ Borisovich Dynkin

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D. Fuchs and A. Schwarz: “Matrix Vieta theorem,” pp. 15–22 in Lie groups and Lie algebras: E. B. Dynkin’s seminar. Edited by S. G. Gindikin and E. B. Vinberg. American Mathematical Society Translations. Series 2 169. American Mathematical Society (Providence, RI), 1995. This book is also no. 26 in the Advances in Mathematics series. MR 1364450 Zbl 0837.15011 ArXiv math/9410207 incollection

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 \[ X^n + A_1 X^{n-1} + \cdots + A_n = 0 ,\] 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.

Evgeniĭ Borisovich Dynkin	Related
Simon Grigorevich Gindikin	Related
Albert S. Schwarz	Related	Volume
Ernest Borisovich Vinberg	Related