Null vectors, Schur complements, and Parter vertices

Fallat, Shaun; Parenteau, Johnna

doi:10.2140/involve.2022.15.843

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Null vectors, Schur complements, and Parter vertices

Shaun Fallat and Johnna Parenteau

Vol. 15 (2022), No. 5, 843–856

DOI: 10.2140/involve.2022.15.843

Abstract

One of the most important historical contributions to the inverse eigenvalue problem associated with trees is the celebrated Parter–Wiener theorem. We offer an alternate elementary proof of this seminal result, utilizing the basic matrix tool known as the Schur-complement of a matrix, in connection with analyzing the nullspace structure of matrices whose graph is a tree.

Keywords

Schur complements, nullspace, Parter vertices, eigenvalues, multiplicity

Mathematical Subject Classification

Primary: 15A18

Secondary: 05C50

Milestones

Received: 12 November 2021

Accepted: 26 January 2022

Published: 3 March 2023

Communicated by Chi-Kwong Li

Authors

Shaun Fallat
Department of Mathematics and Statistics University of Regina SK Canada

Johnna Parenteau
Department of Mathematics and Statistics University of Regina SK Canada

Vol. 15, No. 5, 2022