Topic
Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
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TL;DR: Two algorithms for determining the matrix numerically are proposed in this paper and, besides its easy implementation, offers a new proof of existence because of its global convergence property.
Abstract: Given two vectors $a,\lambda \in R^{n}$, the Schur--Horn theorem states that $a$ majorizes $\lambda$ if and only if there exists a Hermitian matrix $H$ with eigenvalues $\lambda$ and diagonal entries $a$. While the theory is regarded as classical by now, the known proof is not constructive. To construct a Hermitian matrix from its diagonal entries and eigenvalues therefore becomes an interesting and challenging inverse eigenvalue problem. Two algorithms for determining the matrix numerically are proposed in this paper. The lift and projection method is an iterative method that involves an interesting application of the Wielandt--Hoffman theorem. The projected gradient method is a continuous method that, besides its easy implementation, offers a new proof of existence because of its global convergence property.
47 citations
TL;DR: Rudelson and Vershynin this article established the matching upper estimate for the matching matrix with unit variance and suitable moment assumptions, and proved that the smallest singular value s n (A ) is of order n − 1 / 2 with high probability.
46 citations
TL;DR: Linear complementary problems (LCP) were considered in this article, where the authors fixed their notations and considered the solvability of linear complementarity problems with respect to the special properties of the coefficient matrix M.
46 citations
TL;DR: A complex matrix C consisting of a set of perfect sequences is studied, constructed by taking the inverse discrete Fourier transform of a diagonal matrix, in which the diagonal elements comprise an arbitrary periodically perfect sequence gamma.
Abstract: In this paper, a complex matrix C consisting of a set of perfect sequences is studied. The matrix C is constructed by taking the inverse discrete Fourier transform (IDFT) of a diagonal matrix, in which the diagonal elements comprise an arbitrary periodically perfect sequence gamma. Properties of the matrix C are presented. In addition, the Fourier dual E of the matrix C is investigated. When gamma is a Zadoff-Chu sequence for the case of N even, M=1, and g=0, an explicit representation for the matrix E is derived.
46 citations
TL;DR: In this article, a method for calculating the eigenvalues of the transfer matrix is proposed based on some fundamental properties of the R matrix such as unitarity, cross symmetry, and special relations related to the structure of the degeneracy of R matrix at certain points.
Abstract: The method proposed for calculating the eigenvalues of the transfer matrix is based on some fundamental properties of the R matrix such as unitarity, cross symmetry, and special relations related to the structure of the degeneracy of the R matrix at certain points. One can therefore hope that it applies to a large class of models in which the complicated structure of the R matrix makes it impossible to construct n-particle eigenstates of the transfer matrix by the Bethe ansatz.
46 citations