In this paper, the eigenvalue inverse problem for generalized Jacobi matrices with period plus edges is studied, and such problems have some applications in research in the fields of mathematics, engineering, quantum mechanics, and other disciplines. A row-by-row inversion of the nonzero elements of the matrix is carried out with the given two eigenpairs, and several lemmas are formed by discussing the cases during the inversion process, which in turn gives the solution of this inverse problem, and the corresponding existence uniqueness theorem of the solution is obtained. And two numerical examples are used for checking and achieving the expected results.
The inverse problem is a problem as opposed to a positive problem, take a classic example, in mystery novels, the murderer’s crime process can only be traced after the occurrence, and the detection process is a typical inverse problem-solving process. With the development of theory, the Jacobi matrix inverse eigenvalue problem has been widely used in many fields. In this paper, we explore the eigenvalue inverse problem of an anti-tridiagonal matrix with double constraints based on the Jacobi matrix, i.e. an anti-tridiagonal matrix is given one of its eigenpairs, and the submatrix formed by crossing out the first row and the last column of the anti-tridiagonal matrix is given one of its eigenpairs. Then the non-zero elements of the matrix are inverted, and the existence and uniqueness of the solution to the given problem are obtained. In the end, we give two numerical examples to verify the correctness and validity of the solution.
The inverse matrix problem is a hot and active research topic in computational mathematics[1]. It has broad applications in engineering and scientific calculation, and owns a strong background in physics and practical significance[2]. This paper explores the inverse eigenvalue problem of a bordered anti-tridiagonal matrix. It first illustrates the existence and the uniqueness of its solution, the elaborates on the recursive expression of the solution and uses one numerical example to show the effectiveness of the algorithm, and finally concludes that this work is significant and points out suggestions for further study.
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