Extreme spectra realization by real symmetric tridiagonal and real symmetric arrow matrices

Abstract

Abstract. We consider the following two problems: to construct a real symmetric arrow matrix A and to construct a real symmetric tridiagonal matrix A, from a special kind of spectral information: one eigenvalue λ (j) of the j ×j leading principal submatrix A j of A, j = 1, 2, . . . , n; and one eigenpairHere we give a solution to the first problem, introduced in [J. Peng, X.Y. Hu, and L. Zhang. Two inverse eigenvalue problems for a special kind of matrices. Linear Algebra Appl., 416:336-347, 2006.]. In particular, for both problems to have a solution, we give a necessary and sufficient condition in the first case, and a sufficient condition in the second one. In both cases, we also give sufficient conditions in order that the constructed matrices be nonnegative. Our results are constructive and they generate algorithmic procedures to construct such matrices.

On inverse eigenvalue problems for two kinds of special banded matrices

“…n ) is also interlacing realizable by them for any nonnegative number t. Proof. Because from the item (7) we apparently know the items (1), (2), (3), (5) and 6, we only proof the item (7).…”

“…In 2006, Peng [1,2] discussed the inverse eigenvalue problems for tridiagonal matrices and paw form matrices respectively. Then Pickmann [3][4][5] made some refinements to Peng's papers. A remarkable idea in their papers is to construct an n × n matrix with special structure from the given list (λ (1)…”

Inverse eigenproblems of real symmetric doubly arrowhead matrices

“…However, from the practical point of view, only some eigenvalues and/or eigenvectors of the matrix or submatrices can be known (see [2], [6]). In this sense, in [12], the authors consider a special kind of spectral data, this is, the minimal and maximal eigenvalues of all leading principal submatrices of A n , together with an eigenpair of it. Initially, this special kind of spectral data was introduced by Peng et al in [9], and subsequently, it has been considered by several authors (see [4], [8], [10]- [13]).…”

Extreme Spectra Realization by Nonsymmetric Tridiagonal and Nonsymmetric Arrow Matrices

“…, , of a matrix of form (1) or (2), together with an eigenvector of . This type of spectral information has been recently considered in the literature [7][8][9][10]. More precisely, we consider the following problem.…”