Application of Tree-like Structure of Graph to Matrix Analysis

TL;DR

This paper introduces a graph-based approach to matrix analysis, deriving formulas for determinants, eigenvectors, and characteristic polynomials using signless sums over special graphs, aiding in spectral analysis of large sparse matrices.

Contribution

It presents novel signless sum formulas for matrix properties based on graph structures, enhancing spectral analysis especially for sparse and stochastic matrices.

Findings

Formulas for determinants and eigenvectors using graph structures

Signless sums are useful for singular and stochastic matrices

Applicable to spectral analysis of large sparse matrices

Abstract

Formulas for matrix determinants, algebraic adjunctions, characteristic polynomial coefficients, components of eigenvectors are obtained in the form of signless sums of matrix elements products taking by special graphs. Signless formulas are very important for singular and stochastic problems. They are also useful for spectral analysis of large very sparse matrices.