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.

Keywords

Dual Markov chain / Dual numbers / Eigenvalues / Dual primitive matrices / Irreducible nonnegative matrices / 15A66 / 15A18 / 60J10 / 65F15

References

[1] Angeles J. The dual generalized inverses and their applications in kinematic synthesis. Latest Advances in Robot Kinematics, 2012, Dordrecht, Springer112

[2] Berman A, Plemmons RJ. Nonnegative Matrices in the Mathematical Sciences, 1994, Philadelphia, SIAM

[3] Chen, M.-F., Chen, R.-R.: Top eigenpairs of large scale matrices. CSIAM Trans. Appl. Math. 3(1), 1–25 (2022)

[4] Chen M-F, Jia Z-G, Pang H-K. Computing top eigenpairs of Hermitizable matrix. Front. Math. China, 2021, 16: 345-379

[5] Ching, W.K., Fung, E.S., Ng, M.K.: A higher-order Markov model for the Newsboy’s problem. J. Oper. Res. Soc. 54, 291–298 (2003)

[6] Ching, W.K., Huang, X., Ng, M.K., Siu, T.K.: Markov Chains: Models, Algorithms and Applications. Springer, New York (2006)

[7] Cui, C., Qi, L.: A power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix (2023). arXiv:2304.04355

[8] Gu, Y.L., Luh, L.: Dual-number transformation and its applications to robotics. IEEE J. Robot. Autom. 3, 615–623 (1987)

[9] Horn R, Johnson C. Matrix Analysis, 20122Cambridge, Cambridge University Press

[10] Li W, Cui L-B, Ng MK. The perturbation bound for the Perron vector of a transition probability tensor. Numer. Linear Algebra Appl., 2013, 20: 985-1000

[11] Mitrophanov AY. Sensitivity and convergence of uniformly ergodic Markov chains. J. Appl. Probab., 2005, 42(4): 1003-1014

[12] Ng, M., Qi, L., Zhou, G.: Finding the largest eigenvalue of a nonnegative tensor. SIAM J. Matrix Anal. Appl. 31, 1090–1099 (2009)

[13] Pennestri E, Valentini PP. Linear dual algebra algorithms and their applications to kinematics. Multibody Dynamics, 2009, Dordrecht, Springer207229

[14] Pennestri E, Valentini PP, De Falco D, Angeles J. Dual Cayley-Klein parameters and Möbius transform: theory and applications. Mech. Mach. Theory, 2016, 106: 50-67

[15] Qi, L., Alexander, D.M., Chen, Z., Ling, C., Luo, Z.: Low rank approximation of dual complex matrices (2022). arXiv:2201.12781

[16] Qi L, Cui C. Eigenvalues and Jordan forms of dual complex matrices. Commun. Appl. Math. Comput., 2023

[17] Qi L, Ling C, Yan H. Dual quaternions and dual quaternion vectors. Commun. Appl. Math. Comput., 2022, 4: 1494-1508

[18] Qi, L., Luo, Z.: Eigenvalues and singular value decomposition of dual complex matrices (2021). arXiv:2110.02050

[19] Qi L, Luo Z. Eigenvalues and singular values of dual quaternion matrices. Pac. J. Optim., 2023, 19: 257-272

[20] Varga R. Matrix Iterative Analysis, 1962, Englewood Cliffs, Prentice-Hall

[21] Wang H. Characterization and properties of the MPDGI and DMPGI. Mech. Mach. Theory, 2021, 158: 104212

[22] Wang H, Cui C, Wei Y. The QLY least-squares and the QLY least-squares minimal-norm of linear dual least squares problems. Linear Multilinear Algebra, 2023

[23] Wei T, Ding W, Wei Y. Singular value decomposition of dual matrices and its application to traveling wave identification in the brain. J. Matrix Anal. Appl., 2023, 45(1): 634-660 arXiv:2303.01383

[24] Wei Y. Perturbation analysis of singular linear systems with index one. Int. J. Comput. Math., 2000, 74(4483-491

[25] Zhang L, Qi L, Xu Y. Linear convergence of the LZI algorithm for weakly positive tensors. J. Comput. Math., 2012, 30: 24-33

[26] Zhou, J., Wei, Y.: Perturbation analysis of singular linear systems with arbitrary index. Appl. Math. Comput. 145(2/3), 297–305 (2003)

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)

RIGHTS & PERMISSIONS

Shanghai University