Evaluating Matrix Functions by Resummations on Graphs: the Method of Path-Sums

TL;DR

The paper introduces the path-sums method for exactly evaluating functions of matrices with non-commuting entries using graph-based resummation techniques, applicable to finite matrices and various matrix functions.

Contribution

It presents a novel graph-theoretic approach to compute matrix functions and quasideterminants exactly, extending the applicability to non-commuting matrices and complex functions.

Findings

Exact evaluation of matrix functions via path-sums

Path-sums express quasideterminants naturally

Matrix inversion height is NP-complete

Abstract

We introduce the method of path-sums which is a tool for exactly evaluating a function of a discrete matrix with possibly non-commuting entries, based on the closed-form resummation of infinite families of terms in the corresponding Taylor series. If the matrix is finite, our approach yields the exact result in a finite number of steps. We achieve this by combining a mapping between matrix powers and walks on a weighted directed graph with a universal graph-theoretic result on the structure of such walks. We present path-sum expressions for a matrix raised to a complex power, the matrix exponential, matrix inverse, and matrix logarithm. We show that the quasideterminants of a matrix can be naturally formulated in terms of a path-sum, and present examples of the application of the path-sum method. We show that obtaining the inversion height of a matrix inverse and of quasideterminants is an NP-complete problem.