Vieta Matrix and its Determinant

TL;DR

This paper introduces the Vieta matrix, a new matrix constructed from sums and products of variables, and derives a formula for its determinant, expanding understanding of matrix structures related to polynomial roots.

Contribution

The paper defines the Vieta matrix and explicitly calculates its determinant, providing a novel mathematical result related to algebraic structures and polynomial root relationships.

Findings

Determinant formula for the Vieta matrix derived

Vieta matrix links to polynomial root sums and products

Mathematical properties of the Vieta matrix established

Abstract

In this paper, we define a matrix which we call Vieta matrix and calculate its determinant: $$ \left( \begin{array}{cccc} 1&1&\cdots&1\\ a_{2}+a_{3}+\cdots+a_{n}&a_{1}+a_{3}+\cdots+a_{n}&\cdots&a_{1}+a_{2}+\cdots+a_{n-1}\\ a_{2}a_{3}+a_{2}a_{4}+\cdots+a_{n-1}a_{n}&a_{1}a_{3}+a_{1}a_{4}+\cdots+a_{n-1}a_{n}&\cdots&a_{1}a_{2}+a_{1}a_{3}+\cdots+a_{n-2}a_{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ a_{2}a_{3}\dots a_{n-1}a_{n}&a_{1}a_{3}\dots a_{n-1}a_{n}&\cdots&a_{1}a_{2}\dots a_{n-2}a_{n-1} \end{array} \right). $$