Trace identities for skew-symmetric matrices

TL;DR

This paper derives trace identities for skew-symmetric matrices, including formulas for the product of Pfaffians and polynomial expressions for inverse matrices based on traces of powers.

Contribution

It introduces new trace identities relating Pfaffians and inverse matrices of skew-symmetric matrices, expanding theoretical understanding.

Findings

Expressed the product of Pfaffians as a sum involving traces of powers.

Provided polynomial formulas for inverses of skew-symmetric matrices.

Connected inverse matrices to traces of powers of AB.

Abstract

We derive an expression for the product of the Pfaffians of two skew-symmetric matrices A and B as a sum of products of the traces of powers of AB and an expression for the inverse matrix A$^{-1}$, or equivalently B$^{-1}$, as a finite-order polynomial of AB with coefficients depending on the traces of powers of AB.