Open AccessBook
Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory
Dennis S. Bernstein
- 14 Mar 2005
TL;DR: This book brings together a vast body of results on matrix theory for easy reference and immediate application with hundreds of identities, inequalities, and matrix facts stated rigorously and clearly.
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Abstract: "Matrix Mathematics" is a reference work for users of matrices in all branches of engineering, science, and applied mathematics. This book brings together a vast body of results on matrix theory for easy reference and immediate application. Each chapter begins with the development of relevant background theory followed by a large collection of specialized results. Hundreds of identities, inequalities, and matrix facts are stated rigorously and clearly with cross references, citations to the literature, and illuminating remarks. Twelve chapters cover all of the major topics in matrix theory: preliminaries; basic matrix properties; matrix classes and transformations; matrix polynomials and rational transfer functions; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; norms; functions of matrices and their derivatives; the matrix exponential and stability theory; and linear systems and control theory. A detailed list of symbols, a summary of notation and conventions, an extensive bibliography with author index, and an extensive index are provided for ease of use. The book will be useful for students at both the undergraduate and graduate levels, as well as for researchers and practitioners in all branches of engineering, science, and applied mathematics.
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Citations
Weak and Strong Cross Section Dependence and Estimation of Large Panels
TL;DR: In this paper, the authors introduce the concepts of weak and strong cross-section dependence and apply them to the estimation of panel data models with an in-time number of strong and weak common factors.
Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product-Matrix Construction
TL;DR: In this article, the authors presented optimal constructions of minimum bandwidth regenerating (MBR) codes for all values of [n, k, d] and (b) minimum storage regeneration (MSR) code for all value n ≥ 2k-2, using a new product-matrix framework.
Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product-Matrix Construction
TL;DR: To the best of the knowledge, these are the first constructions of exact-regenerating codes that allow the number n of nodes in the network, to be chosen independent of the other parameters.
Large Panels with Common Factors and Spatial Correlation
M. Hashem Pesaran,Elisa Tosetti +1 more
TL;DR: In this paper, the authors consider the statistical analysis of large panel data sets where even after condi- tioning on common observed eects the cross section units might remain dependently distrib- uted.
744
Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry
TL;DR: Two-component spinors are the basic ingredients for describing fermions in quantum field theory in 3 + 1 spacetime dimensions as mentioned in this paper, and they are suitable for practical calculations of crosssections, decay rates, and radiative corrections in the Standard Model and its extensions, including supersymmetry, and many explicit examples are provided.
690
References
Extension of Parrott's theorem to nondefinite scalings
TL;DR: The matrix version of Parrott's theorem is extended to encompass nondefinite scalings, which can be used to relax the causality requirements for many linear matrix inequality based synthesis problems.
10
Fast computation of matrix exponential and logarithm
TL;DR: In this paper, an exponential convergent method for the computation of matrix exponential and logarithm is presented, which is a generalization of the ones given by Borwein El, Brent [3,4] and Newman til] for the scalar case.
10
A Note on Canonical Forms for Matrix Congruence
John M. Lee,David A. Weinberg +1 more
TL;DR: Canonical forms for matrix congruence for general matrices are exhibited as an easy consequence of results presented in a recent paper by R. C. Thompson on pencils of symmetric and skew symmetric matrices as mentioned in this paper.
10
Inequalities on Singular Values of Block Triangular Matrices
Chi-Kwong Li,Roy Mathias +1 more
TL;DR: Using the results, the three questions of Ando on Bloomfield--Watson-type inequalities on eigenvalues are answered and the Kantorovich inequality is generalized.
10
Matrix manifolds and the Jordan structure of the bialternate matrix product
Willy Govaerts,Bart Sijnave +1 more
TL;DR: In this paper, a complete description of the Jordan structure of the bialternate product 2A⊙In of an n×n matrix A is given, which can be used to obtain regular defining systems for some manifolds of matrices which occur naturally in applications, in particular for manifolds with resonant conjugate pairs of pure imaginary eigenvalues.
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