On The Solution Of Yang-baxter-like Matrix Equation For Nilpotent Matrix

When A is a given complex matrix and X is an unknown matrix,the nonlinear matrix equation AX A=XAX is called the Yang-Baxter like matrix equation(YBME).Some scholars have given the structure of YBME solution for some special coefficient matrix A,for example A is a nilpotent matrix,a diagonalizable matrix and a low rank matrix.Some results have been made in the study of the commuting solution of YBME in recent years,but the study of the anti-commuting solution of the equation is still in the initial stage.When the coefficient matrix A is a nilpotent matrix,the structure of the partial anti-commuting solution of YBME is given.This thesis is divided into four chapters and organized as follows:In the first chapter,we briefly introduce the background,research status,existing research results,related theoretical knowledge and the main contents of YBME.In the second chapter,when coefficient matrix A is a nilpotent matrix with index 3,namely A2 ?0,A3=0,the solution method of YBME and the structure of anti-commuting solution are given.Then two numerical examples are given.In the third chapter,we promote and extend the contents of chapter 2.When coefficient matrix A is any nilpotent matrix,namely Az=0,z>1,z ? we give a method to solve the anti-commuting solution of YBME.Then three numerical examples are given.In the last chapter,the main research work of this dissertation is summarized.The further research directions have been proposed.