Topic

# Spectrum of a matrix

About: Spectrum of a matrix is a(n) research topic. Over the lifetime, 1064 publication(s) have been published within this topic receiving 19841 citation(s). The topic is also known as: matrix spectrum.

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13 Dec 2005TL;DR: A theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigen values is proposed.

Abstract: We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a multilinear generalization of the Perron-Frobenius theorem

751 citations

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TL;DR: In this article, the Rayleigh-Ritz variation method for handling linear differential equations is examined and relations between the discrete eigenvalues obtained in successive approximations are established between them.

Abstract: Approximate eigenvalues given by the Rayleigh-Ritz variation method for handling linear differential equations are examined and relations are established between the discrete eigenvalues obtained in successive approximations. These relations should be of use in practical computations. A method for fixing upper bounds to eigenvalues is given and a procedure previously employed by the writer to simplify determinant calculations is adapted for use in the present theory.

515 citations

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TL;DR: The limit of the cumulative distribution function of the eigenvalues is determined by use of a method of moments as discussed by the authors, which is mainly combinatorial, and it is shown that the sum of eigen values, raised to k -th power, k = 1, 2, 3, 4, 5, 6, m is asymptotically normal.

Abstract: Limit theorems are given for the eigenvalues of a sample covariance matrix when the dimension of the matrix as well as the sample size tend to infinity. The limit of the cumulative distribution function of the eigenvalues is determined by use of a method of moments. The proof is mainly combinatorial. By a variant of the method of moments it is shown that the sum of the eigenvalues, raised to k -th power, k = 1, 2,…, m is asymptotically normal. A limit theorem for the log sum of the eigenvalues is completed with estimates of expected value and variance and with bounds of Berry-Esseen type.

384 citations

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TL;DR: This work presents a simple algebraic procedure, based on the Routh-Hurwitz criterion, for determining the character of the eigenvalues without the need for evaluating the eigens explicitly, for a system of nonlinear ordinary differential equations.

Abstract: In stability analysis of nonlinear systems, the character of the eigenvalues of the Jacobian matrix (i.e., whether the real part is positive, negative, or zero) is needed, while the actual value of the eigenvalue is not required. We present a simple algebraic procedure, based on the Routh-Hurwitz criterion, for determining the character of the eigenvalues without the need for evaluating the eigenvalues explicitly. This procedure is illustrated for a system of nonlinear ordinary differential equations we have studied previously. This procedure is simple enough to be used in computer code, and, more importantly, makes the analysis possible even for those cases where the secular equation cannot be solved.

303 citations

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TL;DR: An exact expression for the density of eigenvalues of a random-matrix is derived and it can be seen very directly that it goes over to Wigner's “semi-circle law” when the order of the matrix becomes infinite.

Abstract: An exact expression for the density of eigenvalues of a random-matrix is derived. When the order of the matrix becomes infinite, it can be seen very directly that it goes over to Wigner's “semi-circle law”.

217 citations