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Computing Solutions of the Yang-Baxter-like Matrix Equation for Diagonalisable Matrices

Published online by Cambridge University Press:  06 March 2015

J. Ding  and
Noah H. Rhee
J. Ding*
Affiliation:
Department of Mathematics, The University of Southern Mississippi, USA; School of Mathematical Sciences, Yangzhou University, China
Noah H. Rhee
Affiliation:
Department of Mathematics, University of Missouri – Kansas City, Kansas City, MO 64110-2499, USA
*
*Corresponding author. Email addresses: Jiu.Ding@usm.edu (J. Ding), RheeN@umkc.edu (N. H. Rhee)
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Abstract

The Yang-Baxter-like matrix equation AXA = XAX is reconsidered, where A is any complex square matrix. A collection of spectral solutions for the unknown square matrix X were previously found. When A is diagonalisable, by applying the mean ergodic theorem we propose numerical methods to calculate those solutions.

Keywords

15A2447A35Matrix equationmean ergodic theoremdiagonalisable matrix
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 1 , February 2015 , pp. 75 - 84
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Baxter, R.J., Partition function of the eight-vertex lattice model, Ann. Phys. 70, 193–228 (1972)Google Scholar
[2]Ding, J. and Rhee, N., On the equality of algebraic and geometric multiplicities of matrix eigenvalues, Applied Math Letters 24, 2211–2215 (2011)Google Scholar
[3]Ding, J. and Rhee, N., A nontrivial solution to a stochastic matrix equation, East Asian J. Applied Math. 2, 277–284 (2012)Google Scholar
[4]Ding, J. and Rhee, N., Spectral solutions of the Yang-Baxter-like matrix equation, J. Math. Anal. Appl. 402, 567–573 (2013)Google Scholar
[5]Ding, J., Rhee, N. and Zhang, C., Further solutions of a Yang-Baxter-like matrix equation, East Asian J. Applied Math. 3, 352–362 (2013)Google Scholar
[6]Ding, J. and Zhou, A., Nonnegative Matrices, Positive Operators, and Applications, World Scientific (2009)Google Scholar
[7]Felix, F., Nonlinear Equations, Quantum Groups and Duality Theorems: A Primer on the Yang-Baxter-like Equation, VDM Verlag (2009)Google Scholar
[8]Hillar, C., Levine, L. and Rhea, D., Word equations in a uniquely divisible group, 22nd Int. Conf. on Formal Power Series and Algebraic Combinatorics, Discrete Math. Theor. Comput. Sci. Proc., 749–760, Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2010)Google Scholar
[9]Hietarinta, J., Solving the two-dimensional constant quantum Yang-Baxter-like equation, J. Math. Phys. 34, 1725–1756 (1993)Google Scholar
[10]Jimbo, M. (ed.), Yang-Baxter-like Equation in Integrable Systems, World Scientific (1989)Google Scholar
[11]Meyer, C., Matrix Analysis and Applied Linear Algebra, SIAM (2000)Google Scholar
[12]Sklyanin, E., Some algebraic structures connected with the Yang-Baxter-like equation, Func. Analy. Appl. 16, 263–270 (1982)Google Scholar
[13]Yang, C.N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19, 1312–1315 (1967)Google Scholar
[14]Yang, C. and Ge, M., Braid Group, Knot Theory, and Statistical Mechanics, World Scientific (1989)Google Scholar