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: This technical note considers problems that involve eigenvalues of uncertain matrices expressed in linear fractional form using the structured singular value of LaTeX and results allow information of an uncertain matrix efficiently to be acquired by the existing computational tools.
Abstract: This technical note considers problems that involve eigenvalues of uncertain matrices expressed in linear fractional form using the structured singular value $\mu$ and the skewed structured singular value $
u$ . In particular, positive definiteness conditions, maximum and minimum eigenvalues, and generalized eigenvalues of uncertain matrices are expressed using $\mu$ and $
u$ . The obtained results allow us to acquire information of an uncertain matrix efficiently by the existing computational tools that provide practically useful approximations to the values of $\mu$ and $
u$ .
3 citations
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TL;DR: In this paper, the eigenvalues of a fixed complex matrix and two vectors were studied and the dependence of the intersections on the vectors $u,v$ was studied. But the dependence on the intersections was not considered.
Abstract: Let $A$ be a fixed complex matrix and let $u,v$ be two vectors. The eigenvalues of matrices $A+\tau uv^\top $ $(\tau\in\mathbb{R})$ form a system of intersecting curves. The dependence of the intersections on the vectors $u,v$ is studied.
3 citations
TL;DR: The method is based on the analysis of the norms of vectors whose elements are the normalized eigenvalues of the received signal covariance matrix and the corresponding normalized indexes, allowing for the estimation of the number of sources without the knowledge of any additional parameter.
Abstract: In this paper we propose an empirical method for estimating the number of sources of signals impinging on multiple sensors. The method is based on the analysis of the norms of vectors whose elements are the normalized eigenvalues of the received signal covariance matrix and the corresponding normalized indexes. It is shown that such norms can be used to classify the eigenvalues in two groups: the largest and the remaining ones, thus allowing for the estimation of the number of sources without the knowledge of any additional parameter. It is shown that, in some situations, our norm-based method produces satisfactory performance when compared to a recently proposed random matrix theory method
3 citations
TL;DR: If A is a completely continuous self-adjoint operator on a Hilbert space its eigenvalues are the values of the inner product at stationary points on the unit sphere provided that certain regularity conditions hold at the eigenvectors.
3 citations
01 Jan 1995
TL;DR: A parallel algorithm for nding the singular values of a bidiagonal matrix B by computing the corresponding eigenvalues of the symmetric tridiagonal (ST) matrix B T B and taking the square roots of those eigen values.
Abstract: This paper describes a parallel algorithm for nding the singular values of a bidiagonal matrix B. The algorithm nds the largest singular values by nding the corresponding eigenvalues of the symmetric tridiagonal (ST) matrix B T B and taking the square roots of those eigenvalues. The smallest singular values are calculated by computing the corresponding eigenvalues of another ST matrix T, which contains zeroes in the main diagonal and entries of B in the oo-diagonals. Details of two implementations of the algorithm are described. One implementation uses the split-merge algorithm to nd the eigenvalues of ST matrices, and the other uses a bisection-based eigenvalue method. Performance results on an nCUBE-2 and a workstation cluster are presented.
3 citations