for all complex vectors ai and bj . One can easily prove that if X is positive definite then X is hermitian (see, e.g., Ref. [1], p. 65). Since the eigenvalues of hermitian matrices are real, it is easy to prove that the eigenvalues of positive definite matrices are real and positive. Moreover, a positive definite matrix is invertible, since it does not possess a zero eigenvalue. Note that a no...

In this paper, computational efficient technique is proposed to calculate the eigenvalues of a tridiagonal system matrix using Strum sequence and Gerschgorin theorem. The proposed technique is applicable in various control system and computer engineering applications. KeywordsEigenvalues, tridiagonal matrix, Strum sequence and Gerschgorin theorem. I.INTRODUCTION Solving tridiagonal linear syste...

The spectrum of complex PT-symmetric potential, V (x) = igx, is known to be null. We enclose this potential in a hard-box: V (|x| ≥ 1) = ∞ and in a softbox: V (|x| ≥ 1) = 0. In the former case, we find real discrete spectrum and the exceptional points of the potential. The asymptotic eigenvalues behave as En ∼ n. The solvable purely imaginary PT-symmetric potentials vanishing asymptotically kno...

The spectrum of complex PT-symmetric potential, V (x) = igx, is known to be null. We enclose this potential in a hard-box: V (|x| ≥ 1) = ∞ and in a softbox: V (|x| ≥ 1) = 0. In the former case, we find real discrete spectrum and the exceptional points of the potential. The asymptotic eigenvalues behave as En ∼ n. The solvable purely imaginary PT-symmetric potentials vanishing asymptotically kno...

This paper considers solving the real eigenvalues of the Quadratic Eigenvalue Problem (QEP) Q(λ)x ≡ (λM+λC+K)x = 0 in a given interval (a, b), where the coefficient matrices M , C, K are Hermitian and M is nonsingular. First, an inertia theorem for the QEP is proven, which characterizes the difference of inertia index between Hermitian matrices Q(a) and Q(b). Several useful corollaries are then...

In this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to th...

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler–Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situat...

States with v 1 = λ, v 2 = −λ and reciprocal equations in the six-vertex model Abstract The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of n constants v 1 ,. .. , v n , the zeros of the function Q(v). A peculiar class of eigenvalues are those in which two of the constants v 1 , v 2 are equal to λ, −λ , with ∆ = − cosh λ an...

Flaming phenomena represent the divergence in strength of user dynamics as created by interactions online social networks (OSNs). Although it has been known that flaming occur when Laplacian matrix OSN non-real eigenvalues, was recently shown may even if all eigenvalues are real numbers. This effect appears only situation some degenerate, and a special unitary transformation is applied to equat...

The real ε-pseudospectrum of a real matrix A consists of the eigenvalues of all real matrices that are ε-close in spectral norm to A. The real pseudospectral abscissa, which is the largest real part of these eigenvalues for a prescribed value ε, measures the structured robust stability of A w.r.t. real perturbations. In this report, we introduce a criss-cross type algorithm to compute the real ...