Topic
Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
Papers published on a yearly basis
Papers
More filters
01 Jan 1956
TL;DR: In this paper, it was shown how to define a generalized inverse of a non-singular matrix, which has relevance to the statistical problem of finding the best approximate solution of inconsistent systems of equations by the method of least squares.
Abstract: In an earlier paper (4) it was shown how to define for any matrix a unique generalization of the inverse of a non-singular matrix. The purpose of the present note is to give a further application which has relevance to the statistical problem of finding ‘best’ approximate solutions of inconsistent systems of equations by the method of least squares. Some suggestions for computing this generalized inverse are also given.
718 citations
TL;DR: A theoretical analysis of some Krylov subspace approximations to the matrix exponential operation $\exp (A)v$ is presented, and a priori and a posteriors error estimates are established.
Abstract: In this note a theoretical analysis of some Krylov subspace approximations to the matrix exponential operation $\exp (A)v$ is presented, and a priori and a posteriors error estimates are established. Several such approximations are considered. The main idea of these techniquesis to approximately project the exponential operator onto a small Krylov subspace and to carry out the resulting small exponential matrix computation accurately. This general approach, which has been used with success in several applications, provides a systematic way of defining high-order explicit-type schemes for solving systems of ordinary differential equations or time-dependent partial differential equations.
700 citations
TL;DR: It is shown that properly constrained nuclear-norm minimization stably recovers a low-rank matrix from a constant number of noisy measurements per degree of freedom; this seems to be the first result of this nature.
Abstract: This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear combinations of the matrix entries. We show that properly constrained nuclear-norm minimization stably recovers a low-rank matrix from a constant number of noisy measurements per degree of freedom; this seems to be the first result of this nature. Further, with high probability, the recovery error from noisy data is within a constant of three targets: (1) the minimax risk, (2) an “oracle” error that would be available if the column space of the matrix were known, and (3) a more adaptive “oracle” error which would be available with the knowledge of the column space corresponding to the part of the matrix that stands above the noise. Lastly, the error bounds regarding low-rank matrices are extended to provide an error bound when the matrix has full rank with decaying singular values. The analysis in this paper is based on the restricted isometry property (RIP).
698 citations
Book•
25 Dec 2012
TL;DR: In this article, the second edition of the Second Edition of the first edition, the authors presented a list of symbols for elementary linear and multilinear algebra, including square matrices, tensor and exterior products, with real or complex entries.
Abstract: Preface to the Second Edition.- Preface to the First Edition.- List of Symbols.- 1 Elementary Linear and Multilinear Algebra.- 2 What Are Matrices.- 3 Square Matrices.- 4 Tensor and Exterior Products.- 5 Matrices with Real or Complex Entries.- 6 Hermitian Matrices.- 7 Norms.- 8 Nonnegative Matrices.- 9 Matrices with Entries in a Principal Ideal Domain Jordan Reduction.- 10 Exponential of a Matrix, Polar Decomposition, and Classical Groups.- 11 Matrix Factorizations and Their Applications.- 12 Iterative Methods for Linear Systems.- 13 Approximation of Eigenvalues.- References.- Index of Notation.- General Index.- Cited Names.-
692 citations
01 Feb 1967
TL;DR: A unified treatment of matrix methods useful for field problems is given in this paper, where the basic mathematical concept is the method of moments, by which the functional equations of field theory are reduced to matrix equations.
Abstract: A unified treatment of matrix methods useful for field problems is given. The basic mathematical concept is the method of moments, by which the functional equations of field theory are reduced to matrix equations. Several examples of engineering interest are included to illustrate the procedure. The problem of radiation and scattering by wire objects of arbitrary shape is treated in detail, and illustrative computations are given for linear wires. The wire object is represented by an admittance matrix, and excitation of the object by a voltage matrix. The current on the wire object is given by the product of the admittance matrix with the voltage matrix. Computation of a field quantity corresponds to multiplication of the current matrix by a measurement matrix. These concepts can be generalized to apply to objects of arbitrary geometry and arbitrary material.
669 citations