Trace Formulae of Characteristic Polynomial and Cayley-Hamilton's Theorem, and Applications to Chiral Perturbation Theory and General Relativity

TL;DR

This paper presents new combinatorial proofs for characteristic polynomial recurrence relations, deriving explicit trace formulae for Cayley-Hamilton's theorem and applications to physics theories like chiral perturbation theory and general relativity.

Contribution

It provides a novel combinatorial approach to derive explicit trace formulae and determinant expressions, with applications in theoretical physics.

Findings

Explicit trace formulae for Cayley-Hamilton's theorem.

Complete expression for matrix determinants in terms of traces.

Applications demonstrated in chiral perturbation theory and general relativity.

Abstract

By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and then we obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace formulae of the Cayley-Hamilton's theorem with all coefficients explicitly given, and which implies a byproduct, a complete expression for the determinant of any finite-dimensional matrix in terms of the traces of its successive powers. And we discuss some of their applications to chiral perturbation theory and general relativity.