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Null vectors, Schur complements, and Parter vertices

Shaun Fallat and Johnna Parenteau

Vol. 15 (2022), No. 5, 843–856
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