In the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95, 230201 (2005)], an exact solution was reported for the probability pn,k to find exactly k real eigenvalues in the spectrum of an n × n real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof...

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...

We extend the perturbation theory of Vǐsik, Ljusternik and Lidskĭı for eigenvalues of matrices, using methods of min-plus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulæ, extending the classical ones and solving cases which where singular in previous approaches. Ou...

This paper is concerned with bounds of eigenvalues of a complex matrix. Both lower and upper bounds of modulus of eigenvalues are given by the Stein equation. Furthermore, two sequences are presented which converge to the minimal and the maximal modulus of eigenvalues, respectively. We have to point out that the two sequences are not recommendable for practical use for finding the minimal and t...

Exploiting minmax characterizations for nonlinear and nonoverdamped eigenvalue problems, we prove the existence of a countable set of eigenvalues converging to ∞ and inclusion theorems for a rational spectral problem governing mechanical vibrations of a tube bundle immersed in an incompressible viscous fluid. The paper demonstrates that the variational characterization of eigenvalues is a power...

For a self-adjoint linear operator with discrete spectrum or a Hermitian matrix the “extreme” eigenvalues define the boundaries of clusters in the spectrum of real eigenvalues. The outer extreme ones are the largest and the smallest eigenvalues. If there are extended intervals in the spectrum in which no eigenvalues are present, the eigenvalues bounding these gaps are the inner extreme eigenval...

We investigate the behavior of the Lanczos process when it is used to find all the eigenvalues of large sparse symmetric matrices. We study the convergence of classical Lanczos (i.e., without reorthogonalization) to the point where there is a cluster of Ritz values around each eigenvalue of the input matrix A. At that point, convergence to all the eigenvalues can be ascertained if A has no mult...

In this paper, a novel approach for learning low-dimensional optimized feature space image retrieval with minimum intra-class variance and maximum inter-class is proposed. The classical of Linear Discriminant Analysis (LDA) generally used generating an single-labeled images. Since involves images multiple objects, LDA cannot be directly dimensionality reduction optimization. This problem addres...