Inverse Eigenvalue Problem For A Class Of X-like Matrix

Inverse eigenvalue problem for matrices is an important branch of linear algebra.It is widely used in the field of science.Since twentieth Century 50 years,the first article in this respect published,more and more articles have been published,and acquired many deep and useful results.Today research goal is to construct a number of practical scientific applications require eigenvectors and eigenvalues' s matrices.In this paper,the goal is to construct a class of X-like matrix,thus to study its inverse eigenvalue problem and its inverse problem of generalized eigenvalue.The paper uses simultaneous equations to solve the problem and launch the existence and uniqueness conditions of the problem.The full paper consists of the following four chapters.Chapter One:Introduction.Firstly this part introduces the concept of inverse problems,history.Secondly introduces the research status,difficulties and future applications nowadays about the inverse eigenvalue problem.Finally,the concepts of X-like matrix studied in this paper and its application prospects are introduced.Chapter Two:Inverse eigenvalue problem based on the one kind of X-like.This chapter firstly proposes a class of X-like matrix inverse eigenvalue problem.And pushes the conditions of existence to the matrix,then obtain the expressions of the solution.Based on it,this chapter removes the upper right corner of the X-like elements in the matrix,then obtains a lower triangular matrix of our common,it can also be referred to as degradation of X like matrices. According to the study of inverse eigenvalue problem for X-like's method,this chapter studies inverse eigenvalue problem for such lower triangular matrix,and obtains the expressions of the solution.Chapter Three:Inverse eigenvalue problem for a special kind of degradation of X like matrices.This chapter puts forward on the base of the degradation of X-like matrix in the second chapter,then gets a class of upper triangular matrix,and the relations between the matrix elements according to the equivalence relation,the linear relationship are divided into two categories,each kind of the inverse eigenvalue problems for matrices is studied,then the expressions of the solution are obtained separately.Finally two corresponding numerical examples are provided to check separately.Chapter Four:Generalized inverse eigenvalue problem for a special kind of degradation of X like matrices.This chapter is based on the previous three chapters,the inverse eigenvalue for matrices research is extended to generalized inverse eigenvalue problem for matrices,this chapter studies a class of generalized inverse eigenvalue problem for odd order on upper triangular matrix inverse problem,then the expressions of the solution are given.Finally a numerical example is provided to verify the effectiveness of the algorithm.