A family of determinants associated with a square matrix

TL;DR

This paper introduces an infinite family of determinants linked to a square matrix, with explicit evaluations, motivated by applications in graph spectra, expanding understanding of matrix trace properties.

Contribution

It defines a new family of determinants based on traces of powers of matrices and explicitly evaluates their values, with applications to graph spectral analysis.

Findings

Explicit formulas for determinants of the matrix family.

Connections established between matrix traces and graph spectra.

Potential applications in spectral graph theory.

Abstract

We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our family. The work is motivated by applications to graph spectra.