A New Proof of Sturm's Theorem via Matrix Theory

arXiv2110.15364MaRDI QIDQ6381590

Kaiwen Hou, Bin Li

Publication date: 28 October 2021

Abstract: By the classical Sturm's theorem, the number of distinct real roots of a given real polynomial $f(x)$ within any interval $(a,b]$ can be expressed by the number of variations in the sign of the Sturm chain at the bounds. Through constructing the "Sturm matrix", a symmetric matrix associated with $f(x)$ over $mathbbR[x]$, variations in the sign of $f(x)$ can be characterized by the negative index of inertia. Therefore, this paper offers a new proof of Sturm's theorem using matrix theory.

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