Dual Markov Chain and Dual Number Matrices with Nonnegative Standard Parts

doi:10.1007/s42967-024-00388-9

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (6): 2442-2461.doi: 10.1007/s42967-024-00388-9

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Dual Markov Chain and Dual Number Matrices with Nonnegative Standard Parts

Liqun Qi^1,2, Chunfeng Cui³

1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, Zhejiang, China;
2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China;
3. LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing 100191, China

Received:2023-12-07 Revised:2024-02-11 Published:2025-12-24
Contact: Chunfeng Cui, E-mail:chunfengcui@buaa.edu.cn E-mail:chunfengcui@buaa.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12126608, 12131004), the R &D project of Pazhou Lab (Huangpu) (Grant No. 2023K0603), and the Fundamental Research Funds for the Central Universities (Grant No. YWF-22-T-204).

Abstract

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.

Key words: Dual Markov chain, Dual numbers, Eigenvalues, Dual primitive matrices, Irreducible nonnegative matrices

CLC Number:

Liqun Qi, Chunfeng Cui. Dual Markov Chain and Dual Number Matrices with Nonnegative Standard Parts[J]. Communications on Applied Mathematics and Computation, 2025, 7(6): 2442-2461.

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Dual Markov Chain and Dual Number Matrices with Nonnegative Standard Parts

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