Nullity Invariance for Pivot and the Interlace Polynomial

TL;DR

This paper explores how principal pivot transform affects nullity in matrices and graphs, generalizing the interlace polynomial's recursive relation and providing a simplified proof of its properties.

Contribution

It introduces nullity invariance under principal pivot transform and extends the recursive relation of the interlace polynomial to a broader context.

Findings

Principal pivot transform preserves nullity patterns in matrices.

The recursive relation of the interlace polynomial is generalized.

A simplified proof of the interlace polynomial's properties is provided.

Abstract

We show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given (square) matrix is described by the symmetric difference operator (for sets). We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof.