Inverses of triangular matrices and bipartite graphs

TL;DR

This paper explores the relationship between nonsingular triangular matrices and bipartite graphs, providing combinatorial descriptions of their inverses and conditions for pattern preservation, with applications to weighted trees.

Contribution

It introduces a novel combinatorial approach linking matrix inverses to bipartite graph paths and characterizes matrices with inverse pattern preservation.

Findings

Provides a combinatorial description of matrix inverses using graph paths

Characterizes matrices with inverse pattern preservation under certain conditions

Constructs outer inverses for adjacency matrices of weighted trees

Abstract

To a given nonsingular triangular matrix A with entries from a ring, we associate a weighted bipartite graph G(A) and give a combinatorial description of the inverse of A by employing paths in G(A). Under a certain condition, nonsingular triangular matrices A such that A and A^{-1} have the same zero-nonzero pattern are characterized. A combinatorial construction is given to construct outer inverses of the adjacency matrix of a weighted tree.