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, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion determinant, and a connection formula relating the distributions at the hard and soft edge is obtained, a universal asymptotic behaviour of the two point correlation is identified.
175 citations
172 citations
TL;DR: New mathematical tools, different from those in Baik and Silverstein (2006) [5], are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes.
156 citations
TL;DR: In this article, a deterministic self-adjoint matrix with spectral measure converging to a compactly supported probability measure was perturbed by adding a random finite rank matrix with delocalised eigenvectors and studied the extreme eigenvalues of the deformed model.
Abstract: Consider a deterministic self-adjoint matrix $X_n$ with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalised eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix $X_n$ so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when $X_n$ is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.
149 citations
TL;DR: The coupled channel method for solving the bound-state Schrodinger equation is described in this paper, which is also applicable in other areas of physics, such as particle physics and computer vision.
149 citations