On symmetric matrices with exactly one positive eigenvalue (original) (raw)

On Inclusion and Exclusion Intervals for the Real Eigenvalues of Real Matrices

SIAM Journal on Matrix Analysis and Applications, 2009

Given a real matrix, we analyze an open interval, called row exclusion interval, such that the real eigenvalues do not belong to it. We characterize when the row exclusion interval is nonempty. In addition to the exclusion interval, inclusion intervals for the real eigenvalues, alternative to those provided by the Gerschgorin disks, are also considered for matrices whose offdiagonal entries present a restricted dispersion. The results are applied to obtain a sharp upper bound for the real eigenvalues different from 1 of a positive stochastic matrix and a sufficient condition for the stability of a negative matrix, among other applications.

A note on the convexity of the realizable set of eigenvalues for nonnegative symmetric matrices

Electronic Journal of Linear Algebra, 2001

Geometric properties of the set Rn of n-tuples of realizable spectra of nonnegative symmetric matrices, and the Soules set Sn introduced by McDonald and Neumann, are examined. It is established that S 5 is properly contained in R 5 . Two interesting examples are presented which show that neither Rn nor Sn need be convex. It is proved that Rn and Sn are star convex and centered at .

On Symmetric Nonnegative Matrices with Prescribed Spectrum1

In this paper we give a sufficient condition for the existence and construction of a symmetric nonnegative matrix with prescribed spectrum, and a sufficient conditon for the existence and construction of a 4 × 4 symmetric nonnegative matrix with prescribed spectrum and diagonal entries. This last condition is independent of the sufficient condition given by Fiedler [LAA 9 (1974) 119-142]. We also give some partial answers on an open question of Guo [LAA 266 (1997) 261-270] about symmetric nonnegative matrices.

On symmetric nonnegative matrices with prescribed spectrum

International Mathematical Forum, 2014

In this paper we give a sufficient condition for the existence and construction of a symmetric nonnegative matrix with prescribed spectrum, and a sufficient conditon for the existence and construction of a 4 × 4 symmetric nonnegative matrix with prescribed spectrum and diagonal entries. This last condition is independent of the sufficient condition given by Fiedler [LAA 9 (1974) 119-142]. We also give some partial answers on an open question of Guo [LAA 266 (1997) 261-270] about symmetric nonnegative matrices.

Relaxing the Nonsingularity Assumption for Intervals of Totally Nonnegative Matrices

The Electronic Journal of Linear Algebra, 2020

Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard partial order are considered. It is proven that if the two bound matrices of such a matrix interval are totally nonnegative and satisfy certain conditions, then all matrices from this interval are totally nonnegative and satisfy these conditions, too, hereby relaxing the nonsingularity condition in a former paper [M. Adm, J. Garloff, Intervals of totally nonnegative matrices, Linear Algebra Appl. 439 (2013), pp.3796-3806].

On the positive definiteness and eigenvalues of

2012

In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set S n is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix (S n) f. From this condition we obtain some sufficient conditions for positive definiteness as corollaries. We also use graph theory and show that by making some graph theoretic assumptions on the set S n we are able to reduce the assumptions on the function f while still preserving the positive definiteness of the matrix (S n) f. Dual theorems of these results for join matrices are also presented. As examples we consider the so-called power GCD and power LCM matrices as well as MIN and MAX matrices. Finally we give bounds for the eigenvalues of meet and join matrices in cases when the function f possesses certain monotonic behaviour.

Spectral properties of one class of sign-symmetric matrices

arXiv (Cornell University), 2009

A n × n matrix A, which has a certain sign-symmetric structure (J-signsymmetric), is studied in this paper. It's shown, that such a matrix is similar to a nonnegative matrix. The existence of the second in modulus positive eigenvalue λ 2 of a J-sign-symmetric matrix A, or an odd number k of simple eigenvalues, which coincide with the kth roots of ρ(A) k , is proved under the additional condition, that its second compound matrix is also J-sign-symmetric. The conditions, when a J-sign-symmetric matrix with a J-sign-symmetric second compound matrix has complex eigenvalues, which are equal in modulus to ρ(A), are given.