Solutions Of The Yang-Baxter Matrix Equation For Some Diagonalizable Matrices

The classic Yang-Baxter equation was first introduced by Yang in 1967 and by Baxter independently in 1972.in the field of statistical mechanics.It is now very important in statistical physics and closely related to many fields of mathe-matics such as braid groups and knot theory.The Yang-Baxter matrix equation AX A = XAX is the parameter-free counterpart of the classical Yang-Baxter equa-tion in matrix theory.Usually,such Yang-Baxter matrix equation has infinite many solutions.Although it looks simple in format,Yang-Baxter matrix equation is very difficult to solve for general A since it is equivalent to solving a nonlinear quadratic system of n2 equations with n2 unknown variables.Even in the lower order case of 4×4 matrix,only some concrete and special solutions have been constructed.The research of the Yang-Baxter matrix equation is still in the early stage and how to find out all nontrivial solutions is an important and very challenging problem.Ding and Rhee first studied the Yang-Baxter matrix equation and obtained some important results such as the existence of the solution based on the Brouw-er fixed point theorem and the numerical construction of some solutions via the mean ergodic theorem and direct iteration when A is a nonsingular quasi-stochastic matrix such that A-1 is a stochastic one;the spectral solution based on the spec-tral projectors and generalized eigenspaces.Utilizing the semisimple eigenvalue,Ding and Zhang investigated the structure of spectral solution.For the idempo-tent matrix,Cibotarica,Ding,Kolibal and Rhee used the minimal polynomial and construct implicitly general solutions of the Yang-Baxter matrix equation.Ding and Tian discussed all solutions for a class of elementary matrix with the help of a spectral perturbation result for rank-one updated matrices.Zhou and Chen found all the solutions for matrix of rank two such that zero is its semi-simple eigenvalue of multiplicity n-2.Since finding general solutions of the nonlinear Yang-Baxter matrix equation is difficult,almost all the works so far have been toward constructing commuting solutions of the equation.Ding,Zhang and Rhee discussed the commuting solutions for the nonsingular matrix with some special Jordan forms.By taking advantage of the Jordan form structure of A,together with the help of a well-known theorem on the uniqueness of a solution to Sylvester's equation,Dong and Ding found all the commuting implicit solutions for a diagonalizable matrix.Dong,Ding and Huang used the Jordan block to find all commuting solutions for an arbitrary nilpotent matrix.Dong,Zhou and Chen used the eigenvectors and generalized eigenvectors to construct explicitly the commuting solutions.By a simplified matrix equation,Jordan form and Jordan blocks,Zhou and Ding find all the commuting solutions for a nilpotent matrix of index three.This thesis is devoted to investigate the commuting solutions and general non-commuting solutions with the explicit expression of the Yang-Baxter matrix equa-tion for various types of diagonalizable matrices A.Firstly,by means of the diago-nalization of A and the fact that A has only two distinct eigenvalues,we have found the explicit expression of all solutions for the Yang-Baxter matrix equation when A is a given idempotent matrix or skew idempotent matrix.Secondly,with the help of the theory of Sylvester's equation,similar matrix and matrix blocks,we obtain the characterization of all commuting solutions and then give the explicit expressions of commuting solutions for the Yang-Baxter equatioi of matrix A satisfying A-1 =A or A-1 =-A.Also,we get the explicit expression of all non-commuting solutions under the off-diagonal blocks with a full rank condition and unconditionally.As applications,we give the expressions of non-commuting solutions for the Household matrix.Finally,we consider the more general situation A3 = A or A3 =-A and obtain a different but equivalent form of all solutions for the Yang-Baxter matrix equation via the minimal polynomial,spectral analysis and a spectral perturbation result.In this thesis,we prove that all solutions of Yang-Baxter matrix equation for this type of matrix can be divided into equivalence classes based on similarity,that is,solutions in the same equivalence class are similar.This can bring us more con-venience and new idea to solve some other types of Yang-Baxter matrix equations.Our main results generalize and extend many above-mentioned previously known results.