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 eigenvectors of the inner product-moment of the transformed data matrix are taken directly as the Q-mode scores or scaled by the square roots of their associated eigenvalues and called the R-mode loadings.
Abstract: The dominant feature distinguishing one method of principal components analysis from another is the manner in which the original data are transformed prior to the other computations. The only other distinguishing feature of any importance is whether the eigenvectors of the inner product-moment of the transformed data matrix are taken directly as the Q-mode scores or scaled by the square roots of their associated eigenvalues and called the R-mode loadings. If the eigenvectors are extracted from the product-moment correlation matrix, the variables, in effect, were transformed by column standardization (zero means and unit variances), and the sum of the p-largest eigenvalues divided by the sum of all the eigenvalues indicates the degree to which a model containing pcomponents will account for the total variance in the original data. However, if the data were transformed in any manner other than column standardization, the eigenvalues cannot be used in this manner, but can only be used to determine the degree to which the model will account for the transformed data. Regardless of the type of principal components analysis that is performed—even whether it is Ror Q-mode—the goodness-of-fit of the model to the original data is given better by the eigenvalues of the correlation matrix than by those of the matrix that was actually factored.
39 citations
08 Dec 2008
TL;DR: This paper presents some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of Wishart matrices, using the tensor operator T (.), which was first introduced in .
Abstract: The distribution of the eigenvalues of Wishart matrices and Gaussian quadratic forms is of great interest in communication theory, especially in relation to multiple-input multiple-output (MIMO) systems. In this paper we present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of Wishart matrices, using the tensor operator T (.), which was first introduced in . We obtain both the joint probability distribution function (p.d.f.) of the eigenvalues and the expectation of arbitrary functions of the eigenvalues, including the moments, for the case of both ordered and unordered eigenvalues. These expressions are extremely compact and easy to handle. Application to MIMO systems are discussed.
39 citations
TL;DR: In this article, the authors studied a class of holomorphic matrix models where the integrals are taken over middle-dimensional cycles in the space of complex square matrices and the distribution of eigenvalues is given by a measure with support on a collection of arcs in the complex planes.
39 citations
01 Dec 1987
TL;DR: In this paper, the spectrum of an interval matrix family M is determined by the eigenvalues of the vertices of its vertices, and for the special case of M having a real spectrum, it is shown that this spectrum is completely determined by its eigen values of vertices.
Abstract: In this paper we will examine how the spectrum (of eigenvalues) of an interval matrix family M depends on the spectrum of its extreme sets. We present three results: 1) We show that the roots of pairwise convex combinations of the characteristic polynomials of the vertices of M provide a bound for the spectrum of M. 2) We state conditions on M which guarantee that the spectrum of M can be determined from the spectrum of its relative boundary. 3) For the special case of M having a real spectrum, we show that this spectrum is completely determined by the eigenvalues of its vertices.
39 citations
TL;DR: In this article, the eigenvalue problem for the transfer matrix of the eight-vertex model was studied and the results were used to determine all energy excitations of the XYZ-model.
Abstract: We study the eigenvalue problem for the transfer matrix of the eight-vertex model. By using an inversion relation which was recently discovered we develop a new method to calculate all eigenvalues of the transfer matrix in the thermodynamic limit. This leads to a complete classification of the spectrum. The results are used to determine all energy excitations of theXYZ-model.
37 citations