In this paper, the eigenvalue problem of real symmetric interval matrices is studied. First, in case $2 \times 2$ matrices, all four endpoints two intervals are determined. These not necessarily eigenvalues vertex but it shown that such a matrix can be constructed from original one. Then, necessary and sufficient conditions provided for disjointness intervals. general $n\times n$ case, due to H...

It is shown how pseudo-Hermiticity, a necessary condition satisfied by operators of PT symmetric systems can be introduced in the three Gaussian classes of random matrix theory. The model describes transitions from real eigenvalues to a situation in which, apart from a residual number, the eigenvalues are complex conjugate.

Two different “wave chaotic” systems, involving complex eigenvalues or resonances, can be analyzed using common semiclassical methods. In particular, one obtains fractal Weyl upper bounds for the density of resonances/eigenvalues near the real axis, and a classical dynamical criterion for a spectral gap.

We consider a gyroscopic system with two degrees of freedom under the action of small dissipative and non-conservative positional forces, which has its origin in the models of rotating bodies of revolution being in frictional contact. The spectrum of the unperturbed gyroscopic system forms a ”spectral mesh” in the plane ”frequency – gyroscopic parameter” with double semi-simple purely imaginary...

Much information about the structure and dynamics of a network is encoded in the eigenvalues of its transition matrix. In this paper, we present a first study on the transition matrix of a family of weight driven networks, whose degree, strength, and edge weight obey power-law distributions, as observed in diverse real networks. We analytically obtain all the eigenvalues, as well as their multi...

We study Kostant’s partial order on the elements of a semisimple Lie group in relations with the finite dimensional representations. In particular, we prove the converse statement of [3, Theorem 6.1] on hyperbolic elements. A matrix in GLn(C) is called elliptic (resp. hyperbolic) if it is diagonalizable with norm 1 (resp. real positive) eigenvalues. It is called unipotent if all its eigenvalues...

In this paper we derive bounds on the eigenvalues of the preconditioned matrix that arises in the solution of saddle point problems when the Hermitian and skew-Hermitian splitting preconditioner is employed. We also give sufficient conditions for the eigenvalues to be real. A few numerical experiments are used to illustrate the quality of the bounds.

This paper focuses on the problem of stability in homogeneous systems with dilation. First, we propose an ‘homogeneous eigenvalue’ for homogeneous systems. Next, we analyze the stability of homogeneous systems using homgeneous eigenvalues, and we show that the use of positive real homogeneous eigenvalues implies instability. Finally, we show the effectiveness of the proposed method through an e...