On inverse eigenvalue problems for two kinds of special banded matrices

Abstract

This paper presents two kinds of symmetric tridiagonal plus paw form (hereafter TPPF) matrices, which are the combination of tridiagonal matrices and bordered diagonal matrices. In particular, we exploit the interlacing properties of their eigenvalues. On this basis, the inverse eigenvalue problems for the two kinds of symmetric TPPF matrices are to construct these matrices from the minimal and the maximal eigenvalues of all their leading principal submatrices respectively. The necessary and sufficient conditions for the solvability of the problems are derived. Finally, numerical algorithms and some examples of the results developed here are given.

Eigenvalue inverse problem for generalized Jacobi matrices with periodic plus edges

“…In this paper we discuss the eigenvalue inverse problem for generalized Jacobi matrices with period plus edges, and finally obtain the existence uniqueness theorem for the solution of this problem by performing row-by-row inversion of the matrix, discussing it in several cases, and giving specific expressions for the solution in the case of forming several lemmas. Finally, the effectiveness of the inversion algorithm is verified by numerical examples [7][8][9][10][11][12][13][14] .…”

On the inverse eigenvalue problem for a special kind of acyclic matrices

“…Each path of a generalized star graph is called an arm and the number of vertices of a generalized star graph is the sum of the number of vertices of each arm plus one. Xu and Chen in 2017 [10] investigated matrices of tridiagonal-and-paw and pawand-tridiagonal graphs which we generalized to a larger class of graphs, named generalized star graphs. The graph they studied is shown in Figure 3, labeling the vertices of this graph from left to right or from right to left determines whether it is a tridiagonal-and-paw or a paw-and-tridiagonal graph.…”

“…, n. Sharma and Sen in 2016 [8] and 2018 [9] studied the same problem for symmetric tridiagonal and other kinds of acyclic matrices. In 2017, Xu and Chen [10] solved this problem for two kinds of special banded matrices. In this paper we investigate the same IEP, namely IEPGS, of constructing a special kind of acyclic matrices whose graph is a generalized star graph by the minimal and maximal eigenvalues of its all leading principal submatrices.…”