A formula for the trace of symmetric powers of matrices

TL;DR

This paper derives a formula to compute the trace of symmetric powers of matrices using basic matrix powers, a linear function, and recursively defined polynomials, simplifying calculations in linear algebra.

Contribution

It introduces a novel explicit formula for the trace of symmetric powers of matrices involving recursive polynomial functions and a linear functional.

Findings

Provides a closed-form expression for the trace of symmetric powers

Utilizes recursive polynomial functions for computation

Simplifies calculations of symmetric power traces in linear algebra

Abstract

We present a formula for the trace of any symmetric power of a $n\times n$ matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and $n-2$ polynomial functions defined recursively.