A divide-and-conquer method for constructing a pseudo-Jacobi matrix from mixed given data

Abstract

For the given signature operator H=Ir⊕−In−r, a pseudo-Jacobi matrix is a self-adjoint matrix relatively to a symmetric bilinear form 〈⋅,⋅〉H, and it is the counterpart of a classical Jacobi matrix to the indefinite scalar product space setting. In this article, we consider an inverse eigenvalue problem for this class of matrices. The main concern is to construct an n×n pseudo-Jacobi matrix from a prescribed n-tuple of distinct real numbers and a Jacobi matrix of order not less than ⌊n2⌋, such that its spectrum is this tuple and the given Jacobi matrix is its trailing principal submatrix. A divide-and-conquer scheme is used to solve this problem, and a necessary and sufficient condition under which the problem is solvable is presented. A numerical algorithm is designed to solve this pseudo-Jacobi matrix inverse eigenvalue problem according to the obtained results. Some illustrative numerical examples are also given to test the reconstructive algorithm.