The Nullity Theorem for Principal Pivot Transform

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.

5 pages, no figures

References in corpus (3)

• The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems
• Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices
• Nullity Invariance for Pivot and the Interlace Polynomial