The Inverse Eigenvalue Problem For Anti-Diagonal Matrices

It is called the eigenvalue problem that we want to get the eigenvalue when the matrice has been given, and It is called the inverse eigenvalue problem that we want to get the matrice when the eigenvalue and eigenvalue vector have been given. The inverse eigenvalue problem for matrices has been used frequently in engineering or scientific calculations of practical solutions. It has profound background of physics and practical significance. Many scholars and experts have researched the inverse eigenvalue problem more extensively and in-depth studied, and get lots of results. This paper mainly describes the concepts related to anti-diagonal matrices and has done a thorough study for the inverse eigenvalue problem for anti-diagonal matrices, which are described as follows:In the first part, we introduce the background、the current situation of the inverse eigenvalue problems and the structure of this paper, this paper is being wrote from the from the inverse eigenvalue problems for matrices、the inverse eigenvalue problems for Jacobi matrices to the inverse eigenvalue problem for anti-diagonal matrices which is the most important of this article.The second part, it mainly describes the inverse eigenvalue problem for anti-diagonal matrices, which includes the inverse eigenproblem for anti-tridiagonal matrices、inverse eigenvalue problem for generalized anti-tridiagonal matrices with linear relation and inverse eigenvalue problem for generalized periodic anti-tridiagonal matrices with linear relation. Find out a unique solution of the necessary and sufficient conditions of the problems,give the expression of the solution of the problem, provide by some numerical example.The third part, it mainly describes the inverse eigenvalue problem for anti-five diagonal matrices, find out a unique solution of the necessary and sufficient conditions of the problems,give the expression of the solution of the problem, provide by some numerical example.The four part, it mainly describes the inverse eigenvalue problem for non real symmetric anti-diagonal matrix, which includes the inverse eigenproblem for anti-four diagonal matrices and inverse eigenvalue problem for anti-six diagonal matrices. Find out a unique solution of the necessary and sufficient conditions of the problems, give the expression of the solution of the problem, provide by some numerical example.