Study On Commuting Solutions Of A Special Yang-Baxter-Like Matrix Equation

Yang-Baxter-like matrix equation (YBME) is a non-parametric form of Yang-Baxter equation (YBE) in matrix theory.At presen,many scholars have studied and discussed YBME.The solutions of YBME can be obtained in many ways,such as Brouwer fixed point theorem,mean ergodic theorem and spectral theorem.For some special cases of coefficient matrix A,such as a nonsingular quasi-random matrix,a diagonalizable matrix,an idempotent matrix or similar to a special kind of Jordan-type matrix,researchers have given the structure of all solutions of YBME.However,when coefficient A is an arbitrary matrix,there is still no general method to give all its solutions.Therefore,it is of great significance to further study the structure of its solutions for other special coefficient A.This is the problem to be studied in this paper.Based on the commuting solutions of YBME of coefficient matrix A =PQ~T studied by Ding Jiu et al.,In this paper,we generalize it to the case which satisfies A =I-PQ~T with Q~TP being singular matrix,and give the structure of their commuting solutions.Finally,we give several numerical examples to verify its effectiveness.This thesis is divided into four chapters and organized as follows:In the first chapter,we will introduce the research background,the current research status and the existing research results of YBME.The related theoretical knowledge and the main research contents of this thesis have been presented.In the second chapter,when the coefficient matrix A=I-pq~T,wherepand qare two n dimensional vectors satisfying q~Tp=0,a method of solving YBME commuting solutions and its structure are given,and its effectiveness is illustrated by numerical examples.In the third chapter,when the coefficient matrix A=I-PQ~T,where P and Q are two n×2 dimensional column full rank complex matrices satisfying Q~TP=0,a method of solving YBME commuting solutions and its structure are given.Some numerical examples are given to illustrate its feasibility and effectiveness.In the last chapter,the main research work of this dissertation is summarized,and the further research directions have been proposed.