# American Institute of Mathematical Sciences

2013, 3(2): 347-352. doi: 10.3934/naco.2013.3.347

## Solutions of the Yang-Baxter matrix equation for an idempotent

 1 Department of Mathematics, The Univeristy of Southern Mississippi, Hattiesburg, MS 39406-5045, United States 2 Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406-5045 3 Department of Mathematics, The Univeristy of New Haven, West Haven, CT 06516, United States 4 Department of mathematics and Statistics, The University of Missouri - Kansas City, Kansas City, MO 64110-2499, United States

Received  March 2012 Revised  March 2013 Published  April 2013

Let $A$ be a square matrix which is an idempotent. We find all solutions of the matrix equation of $AXA=XAX$ by using the diagonalization technique for $A$.
Citation: A. Cibotarica, Jiu Ding, J. Kolibal, Noah H. Rhee. Solutions of the Yang-Baxter matrix equation for an idempotent. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 347-352. doi: 10.3934/naco.2013.3.347
##### References:
 [1] R. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain II eqivalence to a generalized ice-type lattice model, Ann Phys., 76 (1973), 25-47. doi: 10.1016/0003-4916(73)90440-5. [2] A. Cibotarica, "An Examination of the Yang-Baxter Equation,'' Master thesis, University of Southern Mississippi in Hattiesberg, 2011. [3] J. Ding and N. Rhee, A nontrivial solution to a stochastic matrix equation, East Asia J. Applied Math., 2 (2012), 277-284. [4] F. Felix, "Nonlinear Equations, Quantum Groups and Duality Theorems: A Primer on the Yang-Baxter Equation," VDM Verlag, 2009. [5] M. Jimbo, "Introduction to the Yang-Baxter equation,'' Braid Group, Knot Theory and Statistical Physics II, World Scientific, (1994), 153-176. [6] C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta function interaction, Phys. Rev. Lett., 19 (1967), 1312-1315. doi: 10.1103/PhysRevLett.19.1312. [7] C. N. Yang and M. L. Ge, "Braid Group, Knot Theory and Statistical Physics II,'' World Scientific, 1994.

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##### References:
 [1] R. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain II eqivalence to a generalized ice-type lattice model, Ann Phys., 76 (1973), 25-47. doi: 10.1016/0003-4916(73)90440-5. [2] A. Cibotarica, "An Examination of the Yang-Baxter Equation,'' Master thesis, University of Southern Mississippi in Hattiesberg, 2011. [3] J. Ding and N. Rhee, A nontrivial solution to a stochastic matrix equation, East Asia J. Applied Math., 2 (2012), 277-284. [4] F. Felix, "Nonlinear Equations, Quantum Groups and Duality Theorems: A Primer on the Yang-Baxter Equation," VDM Verlag, 2009. [5] M. Jimbo, "Introduction to the Yang-Baxter equation,'' Braid Group, Knot Theory and Statistical Physics II, World Scientific, (1994), 153-176. [6] C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta function interaction, Phys. Rev. Lett., 19 (1967), 1312-1315. doi: 10.1103/PhysRevLett.19.1312. [7] C. N. Yang and M. L. Ge, "Braid Group, Knot Theory and Statistical Physics II,'' World Scientific, 1994.
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