Topic
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.
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TL;DR: In this article, it was shown that a certain distance formula, known to be a necessary condition for normality, is in fact sufficient and demonstrates that the spectrum of a matrix can be used to recover the spectral norm of its resolvent precisely when the matrix is normal.
Abstract: Given a normal matrix A and an arbitrary square matrix B (not necessarily of the same size), what relationships between A and B, if any, guarantee that B is also a normal matrix? We provide an answer to this question in terms of pseudospectra and norm behavior. In doing so, we prove that a certain distance formula, known to be a necessary condition for normality, is in fact sufficient and demonstrates that the spectrum of a matrix can be used to recover the spectral norm of its resolvent precisely when the matrix is normal. These results lead to new normality criteria and other interesting consequences.
7 citations
TL;DR: In this article, the Laplace operator acting on the space of automorphic functions with respect to a congruence group is shown to have a point spectrum with Weyl's law and a purely continuous spectrum on [¼,∞] of finite multiplicity equal to the number of inequivalent cusps.
Abstract: Let Δ denote the Laplace operator acting on the space L 2 (Г/ H ) of automorphic functions with respect to a congruence group Г, square integrable over the fundamental domain F =Г/ H . It is known that Δ has a point spectrum with (Weyl's law) and it has a purely continuous spectrum on [¼,∞) of finite multiplicity equal to the number of inequivalent cusps. The eigenpacket of the continuous spectrum is formed by the Eisenstein series E a (z, s) on s = ½+ it where a ranges over inequivalent cusps. The eigenfunctions u i (z) with positive eigenvalues are Maass cusp forms.
7 citations
TL;DR: In this article, the existence of infinitely many negative eigenvalues of a block operator matrix acting in the direct sum of one-and two-particle subspaces of a Fock space is considered.
Abstract: A \(2\times2\) block operator matrix \({\mathbf H}\) acting in the direct sum of one- and two-particle subspaces of a Fock space is considered. The existence of infinitely many negative eigenvalues of \(H_{22}\) (the second diagonal entry of \({\bf H}\)) is proved for the case where both of the associated Friedrichs models have a zero energy resonance. For the number \(N(z)\) of eigenvalues of \(H_{22}\) lying below \(z\lt0\), the following asymptotics is found \[\lim\limits_{z\to -0} N(z) |\log|z||^{-1}=\,{\mathcal U}_0 \quad (0\lt {\mathcal U}_0\lt \infty).\] Under some natural conditions the infiniteness of the number of eigenvalues located respectively inside, in the gap, and below the bottom of the essential spectrum of \({\mathbf H}\) is proved.
7 citations
TL;DR: In this article, the Moore-Penrose inverse of such a "retrocirculant" was determined and the nonzero eigenvalues of the inverse were the reciprocals of the non zero eigen values of the retrocirculant.
7 citations
TL;DR: In this paper, it was shown that for some p there exist small eigenvalues of the Laplacian on p-forms for collapsings of the even dimensional spheres with curvature bounded below.
Abstract: We prove that for some p there exist small eigenvalues of the Laplacian on p-forms for collapsings of the even dimensional spheres with curvature bounded below. These collapsings were constructed by T. Yamaguchi.
7 citations