where the constant C depends on t, γ and d (see also [6] for the case when V is real). The paper [5] deals with the natural question that appears in relation to the main result of [3]: what estimates are valid for the eigenvalues situated inside the conical sector {λ : |Iλ| < tRλ}, where the eigenvalues might be close to the positive half-line? Theorems of the article [5] provide some informati...

Spectral properties of sign symmetric matrices are studied. A criterion for sign symmetry of shifted basic circulant permutation matrices is proven, and is then used to answer the question which complex numbers can serve as eigenvalues of sign symmetric 3 × 3 matrices. The results are applied in the discussion of the eigenvalues of QM -matrices. In particular, it is shown that for every positiv...

Let A be an n x n matrix whose elements are independent randomvariables with standard normal distributions. As n ..... 00 , the expected numberof real eigenvalues is asymptotic to .J2nln. We obtain a closed form expres-sion for the expected number of real eigenvalues for finite n, and a formula forthe density of a real eigenvalue for finite n. Asymptotically, a real normalized<l...

We present an elementary proof that the qualitative picture of a linear hyperbolic flow is insensitive to slight measurements errors in its associated vector field. 1 Robustness of the linear hyperbolic flows We start considering linear flows e , where the matrix H is hyperbolic, i.e., all its eigenvalues have nonzero real part. The set H of all hyperbolic matrices is generic, in the sense that...

We define a word in two positive definite (complex Hermitian) matrices A and B as a finite product of real powers of A and B. The question of which words have only positive eigenvalues is addressed. This question was raised some time ago in connection with a long-standing problem in theoretical physics, and it was previously approached by the authors for words in two real positive definite matr...

If a Sturm-Liouville problem is given in an open interval of the real line then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues a...

We study the stability of the replica-symmetric solution of a two-sublattice infinite-range spin-glass model, which can describe the transition from an antiferromagnetic to a spin-glass state. The eigenvalues associated with replica-symmetric perturbations are in general complex. The natural generalization of the usual stability condition is to require the real part of these eigenvalues to be p...

We study the eigenvalues of time-harmonic Maxwell's equations in a cavity upon changes electric permittivity $\varepsilon$ medium. prove that all eigenvalues, both simple and multiple, are locally Lipschitz continuous with respect to $\varepsilon$. Next, we show symmetric functions multiple depend real analytically provide an explicit formula for their derivative As application these results, g...

For every 2n×2n real positive definite matrix A, there exists a symplectic M such that MTAM=diag(D,D), where D is the n×n diagonal with entries d1(A)≤⋯≤dn(A). The numbers d1(A),…,dn(A) are called eigenvalues of A. We derive analogues Wielandt's extremal principle and multiplicative Lidskii's inequalities for eigenvalues.

One of the popular test matrices for eigenvalue routines is Frank matrix due to its well-conditioned and poorly conditioned eigenvalues. All eigenvalues are real, positive different. Sturm Theorem a very useful tool computing tridiagonal symmetric matrices. In this paper, we apply generalized which special form Hessenberg examine by using property. Moreover, illustrate our results with an examp...