Dual Markov Chain and Dual Number Matrices with Nonnegative Standard Parts

Abstract

We propose a dual Markov chain model to accommodate probabilities as well as perturbation, error bounds, or variances, in the Markov chain process. This motivates us to extend the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts. It is shown 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. An explicit formula to compute the dual part of this positive dual number eigenvalue is presented. 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. An upper bound on the distance of stationary states between the dual Markov chain and the perturbed Markov chain is given. Numerical results on both synthetic examples and the dual Markov chain including some real world examples are reported.

Funding

Natural Science Foundation of China(12126608, 12131004)

R &D project of Pazhou Lab (Huangpu)(2023K0603)

Fundamental Research Funds for Central Universities of the Central South University(YWF-22-T-204)