New Results On The Matrix Eigenvector-Eigenvalue-Identity And The Yang-Baxter Matrix Equation

For Hermite matrix and normal matrix,the relation between the square of component module of eigenvector of matrix and eigenvalue satisfies the following relation:(?)The main work of this paper is to find out the similar relation for more general matrices.The matrix equation AXA=XAX is called the Yang-Baxter matrix equation because it is similar in form to the classical Yang-Baxter equation.Although the study on the commuting and non-commuting solutions of the Yang-Baxter matrix equation has achieved fruitful results when the matrix A satisfies certain conditions,there is no general method to solve the Yang-Baxter matrix equation up to now.Therefore,for different types of matrices A,finding out all solutions of Yang-Baxter matrix equation has important research significance and application value.In this paper,the following main works are given.1.When A is a nilpotent matrix of index 2,that is,A2=0,the structure of all solutions to the YangB axter matrix equation is given.2.In the case that A is a diagonalizable matrix with three different eigenvalues 0,?,and ?,all the visible solutions of the Yang-Baxter matrix equation are given.