Spectral properties of one class of sign-symmertic 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.

A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named J--sign-symmetric. The spectrum of a J--sign-symmetric irreducible matrix is studied under assumptions that its second compound matrix is also J--sign-symmetric and irreducible. The conditions, when such matrices have complex eigenvalues on the largest spectral circle, are given. The existence of two positive simple eigenvalues $\lambda_1 > \lambda_2 > 0$ of a J--sign-symmetric irreducible matrix A is proved under some additional conditions. The question, when the approximation of a J--sign-symmetric matrix with a J--sign-symmetric second compound matrix by strictly J--sign-symmetric matrices with strictly J--sign-symmetric compound matrices is possible, is also studied in this paper. Comment: 24 pages

Http Dx Doi Org 10 1080 03081087 2010 547495, 2011

For a square (0, 1, −1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this paper, we investigate the relationship between sign patterns and matrices that diagonalise an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalises some irreducible nonnegative matrix. Further, we characterise the sign patterns S such that some member of Q(S) diagonalises an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realised as the spectrum of an irreducible nonnegative matrix M that is diagonalised by a matrix in the qualitative class of some S 2 N S sign pattern.

Linear Multilinear Algebra, 1996

We investigate matrices which have a positive eigenvalue by virtue of their sign{pattern and regardless of the magnitudes of the entries. When all the o {diagonal entries are nonzero, we show that an n n sign{pattern, n 6 = 3; 4, requires a positive eigenvalue if and only if it has at least one nonnegative diagonal entry and every cycle of length greater than one in its signed digraph is positive. When n = 3; 4, or when not all o {diagonal entries are nonzero, positivity of the cycles of length greater than one is no longer necessary. In the course of proving these results we observe certain necessary and certain su cient

Algebra and Discrete Mathematics

For a complex matrix M, we denote by Sp(M) the spectrum of M and by |M| its absolute value, that is the matrix obtained from M by replacing each entry of M by its absolute value. Let A be a nonnegative real matrix, we call a signing of A every real matrix B such that |B|=A. In this paper, we characterize the set of all signings of A such that Sp(B)=αSp(A) where α is a complex unit number. Our motivation comes from some recent results about the relationship between the spectrum of a graph and the skew spectra of its orientations.

Linear Algebra and its Applications, 1995

A real matrix A is nearly sign-nonsingular if every term in the expansion of det A but one has the same sign. We show such matrices can be put into a normal form in which all diagonal entries are negative, all other nonzero entries are positive, and the directed graph of the matrix is intercyclic. With the help of recent

Linear Algebra and its Applications, 1996

Communications of the Korean Mathematical Society

A real m n matrix A is semipositive (SP) if there is a vector x 0 such that Ax > 0, inequalities being entrywise. A is minimally semipositive (MSP) i f A is semipositive and no column deleted submatrix of A is semipositive. We give a necessary and suucient condition for the sign pattern matrix with n positive e n tries to be minimally semipositive.

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.

Electronic Journal of Linear Algebra, 2010

We present a class of nonsingular matrices, the M C ′ -matrices, and prove that the class of symmetric M C-matrices introduced by Shen, Huang and Jing [On inclusion and exclusion intervals for the real eigenvalues of real matrices. SIAM J. Matrix Anal. Appl., 31:816-830, 2009] and the class of symmetric M C ′ -matrices are both subsets of the class of symmetric matrices with exactly one positive eigenvalue. Some other sufficient conditions for a symmetric matrix to have exactly one positive eigenvalue are derived.