Abstract
In this paper, we study the stochastic root matrices of stochastic matrices. All stochastic roots of 2×2 stochastic matrices are found explicitly. A method based on characteristic polynomial of matrix is developed to find all real root matrices that are functions of the original 3×3 matrix, including all possible (function) stochastic root matrices. In addition, we comment on some numerical methods for computing stochastic root matrices of stochastic matrices.
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References
Bapat, R.B. and T.E.S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, Cambridge, 1996.
Bertsekas, D.P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.
Bhatia, R., Matrix Analysis, Springer Verlag, New York, 1997.
Bjorck, A. and Hammarling, S., “A Schur method for the square root of a matrix”, Linear Algebra and Its Applications, Vol. 52/53, pp127–140, 1983.
Denman, E.D., “Roots of real matrices”, Linear Algebra and Its Applications, Vol. 36, pp133–139, 1981.
Egilsson, T., “Effects of Queueing Congestion on Runway Capacity at Airports”, Master Thesis, Department of Industrial Engineering, Dalhousie University, Halifax, Canada, 2001.
Elfving, G., “Zur Theorie der Markoffschen Ketten”, Acta Soc. Sci. Fennicae n. Ser. A 2, No. 8, pp2–17, 1937.
Higham, N.J., “Newton’s method for the matrix square root”, Mathematics of Computation, Vol. 46, No. 174, pp537–549, 1986.
Higham, N.J., “Computing real square roots of a real matrix”, Linear Algebra and its applications, Vol. 88/89, pp405–430, 1987.
Higham, N.J., Accuracy and Stability of Numerical Algorithms, 2nd Edition, SIAM, Philadelphia, USA, 2002.
Kingman, J.F.C., “The imbedding problem for finite Markov chains”, Z. Wahrscheinlichkeitstheorie, Vol. 1, pp14–24, 1962.
Lancaster, P. and M. Tismenetsky, The Theory of Matrices, Academic, New York, 1985.
Minc, H., Nonnegative Matrices, Wiley, New York, 1988.
Neuts, M.F., Probability, Allyn and Bacon, Boston, Massachusetts, 1973.
Seneta, E., Non-negative Matrices: An Introduction to Theory and Applications, John Wiley & Sons, New York, 1973.
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Qi-Ming He is currently an associate professor in the Industrial Engineering Department at Dalhousie University. He received a Ph.D from the Institute of Applied Mathematics, Chinese Academy of Sciences in 1989 and a Ph.D from the Department of Management Science at the University of Waterloo in 1996. His main research areas are algorithmic methods in applied probability, queuing theory, inventory control, and production management. In investigating various stochastic models, his favourite methods are matrix analytic methods. Recently, he is working on queuing systems with multiple types of customers and inventory systems with multiple types of demands.
Eldon Gunn is currently a professor in the Industrial Engineering Department at Dalhousie University. He received a Ph.D from the University of Toronto in 1981. His research involves the application of operations research methods to production planning, particularly in the natural resource sector. His other research interest includes stochastic programming, routing and scheduling, control of queuing processes and inventory control.
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He, QM., Gunn, E. A note on the stochastic roots of stochastic matrices. J. Syst. Sci. Syst. Eng. 12, 210–223 (2003). https://doi.org/10.1007/s11518-006-0131-9
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DOI: https://doi.org/10.1007/s11518-006-0131-9