The determinant of the second additive compound of a square matrix: a formula and applications

TL;DR

This paper introduces a formula for calculating the determinant of the second additive compound of a square matrix using its characteristic polynomial coefficients, enabling analysis of eigenvalues and sign patterns.

Contribution

It provides a novel explicit formula linking the second additive compound determinant to characteristic polynomial coefficients, with applications to eigenvalue analysis.

Findings

Derived a new formula for the second additive compound determinant

Applied the formula to analyze eigenvalues of polynomial matrices

Extended the formula to cases with sign pattern constraints

Abstract

A formula is presented for the determinant of the second additive compound of a square matrix in terms of coefficients of its characteristic polynomial. This formula can be used to make claims about the eigenvalues of polynomial matrices, with sign patterns as an important special case. A number of corollaries and applications of this formula are given.