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A Nontrivial Solution to a Stochastic Matrix Equation

Published online by Cambridge University Press:  28 May 2015

J. Ding  and
N. H. Rhee
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: [email protected]
Corresponding author. Email: [email protected]
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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.

Keywords

Matrix equationBrouwer's fixed point theorem
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 4 , November 2012 , pp. 277 - 284
Copyright
Copyright © Global-Science Press 2012

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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), 673–691.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), 151–167.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), 1042–1048.CrossRefGoogle Scholar