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.

Iterative Methods for Transmission Eigenvalues

Abstract: Transmission eigenvalues have important applications in inverse scattering theory. They can be used to obtain useful information of the physical properties, such as the index of refraction, of the scattering target. Despite considerable effort devoted to the existence and estimation for the transmission eigenvalues, the numerical treatment is limited. Since the problem is nonstandard, classical finite element methods result in non-Hermitian matrix eigenvalue problems. In this paper, we focus on the computation of a few lowest transmission eigenvalues which are of practical importance. Instead of a non-Hermitian problem, we work on a series of generalized Hermitian problems. We first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem. The roots of these functions are the transmission eigenvalues. Then we apply iterative methods to compute the transmission eigenvalues. We show the convergence of the numerical schemes. The effectiveness of the methods is demonstrated using various numerical examples.

Combining eigenvalues and variation of eigenvectors for order determination

Abstract: In applying statistical methods such as principal component analysis, canonical correlation analysis, and sufficient dimension reduction, we need to determine how many eigenvectors of a random matrix are important for estimation. This problem is known as order determination, and amounts to estimating the rank of a matrix. Previous order-determination procedures rely either on the decreasing pattern, or elbow, of the eigenvalues, or on the increasing pattern of the variability in the directions of the eigenvectors. In this paper we propose a new order-determination procedure by exploiting both patterns: when the eigenvalues of a random matrix are close together, their eigenvectors tend to vary greatly; when the eigenvalues are far apart, their variability tends to be small. The combination of both helps to pinpoint the rank of a matrix more precisely than the previous methods. We establish the consistency of the new order-determination procedure, and compare it with other such procedures by simulation and in an applied setting.

Eigenvalues of the stability matrix for Parisi solution of the long-range spin-glass

Abstract: We study, near ${T}_{c}$, the stability of Parisi's solution for the long-range spin-glass. In addition to the discrete, "longitudinal" spectrum found by Thouless, de Almeida, and Kosterlitz, we find "transverse" bands depending on one or two continuous parameters, and a host of zero modes occupying most of the parameter space. All eigenvalues are non-negative, proving that Parisi's solution is marginally stable.

Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians

Abstract: We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent: (1) H is pseudo-Hermitian; (2) the spectrum of H consists of real and/or complex-conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex-conjugate eigenvalues are identical; (3) H is Hermitian with respect to a positive-semidefinite inner product. We further discuss the relevance of our findings for the merging of a complex-conjugate pair of eigenvalues of diagonalizable pseudo-Hermitian Hamiltonians in general, and the PT-symmetric Hamiltonians and the effective Hamiltonian for a certain closed FRW minisuperspace quantum cosmological model in particular.

Eigenvalues for a class of homogeneous cone maps arising from max-plus operators

Abstract: We study the nonlinear eigenvalue problem $f(x) = \lambda x$ for a class of maps $f: K\to K$ which are homogeneous of degree one and order-preserving, where $K\subset X$ is a closed convex cone in a Banach space X. Solutions are obtained, in part, using a theory of the "cone spectral radius" which we develop. Principal technical tools are the generalized measure of noncompactness and related degree-theoretic techniques. We apply our results to a class of problems max $\max_{t\in J(s)} a(s, t)x(t) = \lambda x(s)$ arising from so-called "max-plus operators," where we seek a nonnegative eigenfunction $ x\in C[0, \mu]$ and eigenvalue $\lambda$. Here $J(s) = [\alpha(s), \beta(s)] \subset [0, \mu]$ for $s\in [0, \mu]$, with $a, \alpha$, and $\beta$ given functions, and the function $a$ nonnegative.