Graph explorer

Matrix Vieta Theorem

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.