The Nullity Theorem for Principal Pivot Transform

TL;DR

This paper extends the nullity theorem from matrix inversion to principal pivot transform, demonstrating invariance of the nullity polynomial and connecting to graph local complementation.

Contribution

It generalizes the nullity theorem to principal pivot transform and shows invariance of the nullity polynomial under this operation.

Findings

Nullity polynomial is invariant under principal pivot transform

Generalization of nullity theorem from matrix inversion to pivot transform

Connections to local complementation on graphs

Abstract

We generalize the nullity theorem of Gustafson [Linear Algebra Appl. (1984)] from matrix inversion to principal pivot transform. Several special cases of the obtained result are known in the literature, such as a result concerning local complementation on graphs. As an application, we show that a particular matrix polynomial, the so-called nullity polynomial, is invariant under principal pivot transform.