Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

Open set of eigenvalues and SVEP

Abstract: The single-valued extension property (SVEP) plays an important role in the local spectral theory. A bounded linear operator T acting on a complex Banach space X is said to have SVEP if there is no analytic function f : U → X defined on a nonempty open subset U of the complex plane such that f 6≡ 0 and (T − z)f(z) ≡ 0 (z ∈ U). By definition, every operator without SVEP has a nonempty open set consisting of eigenvalues. So if the point spectrum of an operator has empty interior, then the operator has automatically SVEP. On the other hand, it is easy to construct an operator T with SVEP such that the interior of the point spectrum σp(T ) is nonempty. As a simple example, see [LN], p. 15, let D := {z ∈ C; |z| < 1} be the open unit disc, let X be the Banach space of all bounded complex-valued functions on D with the supremum norm and let T ∈ B(X) be the operator of multiplication by the independent variable z. It is easy to see that T has SVEP and σp(T ) = D. Moreover, T is decomposable. Note that the Banach space in the last example is non-separable. At first glance it seems that a separable Banach space is too ”small” for the existence of an operator T with SVEP and with nonempty interior of the point spectrum. This motivates the following question which was raised at the Workshop on Operator Theory in Warsaw, 2004.

Density profiles of small Dirac operator eigenvalues for two color QCD at nonzero chemical potential compared to matrix models

Abstract: We investigate the eigenvalue spectrum of the stagerred Dirac matrix in two color QCD at finite chemical potential. The profiles of complex eigenvalues close to the origin are compared to a complex generalization of the chiral Gaussian Symplectic Ensemble, confirming its predictions for weak and strong non-Hermiticity. They differ from the QCD symmetry class with three colors by a level repulsion from both the real and imaginary axis.

Resonance Problems Which Intersect Many Eigenvalues

Abstract: We study semilinear problems in which the nonlinear part is linear at infinity but behaves like a function intersecting many of the eigenvalues of the linear operator. The asymptotic linear operator can have a null space of any finite dimension.

Simplified Eigenvalues Distributions of 2 × 2 Complex Noncentral Wishart

Abstract: This paper derives new simplified analytical cumulative density functions for the eigenvalues of complex noncentral Wishart matrix of size 2times2 Such distributions are often encountered in multiple input multiple output (MIMO) Ricean channels The results derived herein are general for any arbitrary non-centrality matrix, and they account the cases having identical or non-identical eigenvalues of the underlying non-centrality matrix When compared to the generalized distribution recently found in literature which only treated the case of non-identical eigenvalue of the non-centrality matrix, the expressions presented in this paper exhibit a larger parameters range to which the numerical calculation remains computationally viable

On the distribution of the eigenvalues of a matrix differential operator

Abstract: The paper deals with the nature of the spectrum associated with the type of second-ordcr matrix differential operator with catain boundary conditions. It is found that under certain conditions. satisfied by the co-efficients of the differential system, the spectrum is discrete. Some results are then obtained giving distributions of the eigenvalues on the real axis. The method employed depends, among others upon some of the ideas and techniques of E. C. Titchmarsh.