Dual Number Matrices with Primitive and Irreducible Nonnegative Standard Parts

TL;DR

This paper extends Perron-Frobenius theory to dual number matrices with nonnegative parts, providing eigenvalue properties, an explicit eigenvalue formula, an algorithm, and applications to dual Markov chains.

Contribution

It introduces a novel extension of Perron-Frobenius theory to dual number matrices with positive standard parts, including eigenvalue analysis and an algorithm for dual Markov chains.

Findings

Positive dual number matrices have a positive eigenvalue with a positive eigenvector.

The standard part of the dominant eigenvalue exceeds or equals the modulus of others.

An algorithm based on Collatz minimax theorem converges and bounds stationary state differences.

Abstract

In this paper, we extend the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts. One motivation of our research is to consider probabilities as well as perturbation, or error bounds, or variances, in the Markov chain process. We show that such a dual number matrix always has a positive dual number eigenvalue with a positive dual number eigenvector. The standard part of this positive dual number eigenvalue is larger than or equal to the modulus of the standard part of any other eigenvalue of this dual number matrix. We present an explicit formula to compute the dual part of this positive dual number eigenvalue. The Collatz minimax theorem also holds here.The results are nontrivial as even a positive dual number matrix may have no eigenvalue at all. An algorithm based upon the Collatz minimax theorem is constructed. The convergence of the algorithm is studied. We show the upper bounds on the distance of stationary states between the dual Markov chain and the perturbed Markov chain. Numerical results on both synthetic examples and dual Markov chain including some real world examples are reported.