Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-18T01:22:33.939Z Has data issue: false hasContentIssue false

A Nontrivial Solution to a Stochastic Matrix Equation

Published online by Cambridge University Press:  28 May 2015

J. Ding*
Affiliation:
Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406-5045, USA
N. H. Rhee*
Affiliation:
Department of Mathematics and Statistics, University of Missouri - Kansas City, Kansas City, MO 64110-2499, USA
*
Corresponding author. Email: Jiu.Ding@usm.edu
Corresponding author. Email: rheen@umkc.edu
Get access

Abstract.

If A is a nonsingular matrix such that its inverse is a stochastic matrix, the classic Brouwer fixed point theorem implies that the matrix equation AXA = XAX has a nontrivial solution. An explicit expression of this nontrivial solution is found via the mean ergodic theorem, and fixed point iteration is considered to find a nontrivial solution.

Type
Research Article
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Cullen, C., Matrices and Linear Transformations, Dover, 1990.Google Scholar
[2]Ding, J. and Rhee, N., When a matrix and its inverse are stochastic, College Math. J., to appear.Google Scholar
[3]Ding, J. and Zhou, A., Nonnegative Matrices, Positive Operators, and Applications, World Scientific, 2009.CrossRefGoogle Scholar
[4]Felix, F., Nonlinear Equations, Quantum Groups and Duality Theorems: A Primer on the Yang-Baxter Equation, VDM Verlag, 2009.Google Scholar
[5]He, C., Meini, B., and Rhee, N., A shifted cyclic reduction algorithm for quasi-birth-death problems, SIAM J. Matrix Anal. Appl. 23 (2002), 673691.CrossRefGoogle Scholar
[6]He, C., Meini, B., Rhee, N., and Sohraby, K., A quadratically convergent Bernoulli-like algorithm for solving matrix polynomial equation in Markov chains, Elec. Trans. Numer. Anal. 17 (2004), 151167.Google Scholar
[7]Meyer, C., Matrix Analysis and Applied Linear Algebra, SIAM, 2000.Google Scholar
[8]Rhee, N., Note on functional iteration technique for M/G/1 type Markov chains, Linear Algb. Its Appl. 432 (2010), 10421048.CrossRefGoogle Scholar