On the Moore-Penrose pseudo-inversion of block symmetric matrices and its application in the graph theory

• Published in: Linear algebra and its applications Vol. 673; pp. 280 - 303
• Main Authors: Pavlíková, Soňa, Ševčovič, Daniel
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.09.2023
• Subjects: Bridged graph generalized Schur complement; Positively and negatively pseudo-invertibility; Pseudo-invertible matrix and graph; 05C22; 05C50; Positively and negatively pseudo-invertibility; Bridged graph generalized Schur complement; 05B20; Pseudo-invertible matrix and graph; 15B36; 15A09; 15A18
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2023.05.016

The purpose of this paper is to analyze the Moore-Penrose pseudo-inversion of symmetric real matrices with application in the graph theory. We introduce a novel concept of positively and negatively pseudo-inverse matrices and graphs. We also give sufficient conditions on the elements of a block symmetric matrix yielding an explicit form of its Moore-Penrose pseudo-inversion. Using the explicit form of the pseudo-inverse matrix we can construct pseudo-inverse graphs for a class of graphs which are constructed from the original graph by adding pendent vertices or pendant paths.