A Combinatorial Interpretation of the LDU Decomposition of Totally Positive Matrices

TL;DR

This paper provides a combinatorial framework for understanding the LDU decomposition of totally positive matrices, including descriptions of the factors and their inverses, along with recursive computation formulas.

Contribution

It introduces a novel combinatorial interpretation of the LDU decomposition for totally positive matrices, extending existing theories with explicit descriptions and recursive formulas.

Findings

Combinatorial descriptions of L, D, U matrices in LDU decomposition

Explicit formulas for inverses of these matrices

Recursive algorithms for computing L, D, U matrices

Abstract

We study the combinatorial description of the LDU decomposition of totally positive matrices. We give a description of the lower triangular L, the diagonal D, and the upper triangular U matrices of the LDU decomposition of totally positive matrices in terms of the combinatorial structure of essential planar networks described by Zelvinsky and Fomin. Similarly, we find a combinatorial description of the inverses of these matrices. In addition, we provide recursive formulae for computing the L, D, and U matrices of a totally positive matrix.