Generalized vectorization, cross-products, and matrix calculus

Darrell A. Turkington

This book presents the reader with new operators and matrices that arise in the area of matrix calculus. The properties of these mathematical concepts are investigated and linked with zero-one matrices such as the commutation matrix. Elimination and duplication matrices are revisited and partitioned into submatrices. Studying the properties of these submatrices facilitates achieving new results for the original matrices themselves. Different concepts of matrix derivatives are presented and transformation principles linking these concepts are obtained. One of these concepts is used to derive new matrix calculus results, some involving the new operators and others the derivatives of the operators themselves. The last chapter contains applications of matrix calculus, including optimization, differentiation of log-likelihood functions, iterative interpretations of maximum likelihood estimators and a Lagrangian multiplier test for endogeneity

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Alternative title:	Generalized Vectorization, Cross-Products, & Matrix Calculus
Year of publication:	2013 ; First published
Authors:	Turkington, Darrell A.
Publisher:	Cambridge : Cambridge University Press
Subject:	Lineare Algebra \| Linear algebra \| Matrix <Mathematik> \| Vektoranalysis
Description of contents:	1. Mathematical prerequisites -- 2. Zero-one matrices -- 3. Elimination and duplication matrices -- 4. Matrix calculus -- 5. New matrix calculus results -- 6. Applications Table of Contents [gbv.de] ; Description [zbmath.org]

Extent:	1 Online-Ressource (XI, 267 Seiten)
Series:	Cambridge books online
Type of publication:	Book / Working Paper
Language:	English
Notes:	Title from publisher's bibliographic system (viewed on 24 Feb 2016) Systemvoraussetzungen: Internet-Zugriff, Adobe Acrobat Reader Contents; Preface; one Mathematical Prerequisites; 1.1 Introduction; 1.2 Kronecker Products; 1.3 Cross-Product of Matrices; 1.4 Vecs, Rvecs, Generalized Vecs, and Rvecs; 1.4.1 Basic Operators; 1.4.2 Vecs, Rvecs, and the Cross-Product Operator; 1.4.3 Related Operators: Vech and; 1.4.4 Generalized Vecs and Generalized Rvecs; 1.4.5 Generalized Vec Operators and the Cross-Product Operator; two Zero-One Matrices; 2.1 Introduction; 2.2 Selection Matrices and Permutation Matrices; 2.3 The Elementary Matrix; 2.4 The Commutation Matrix; 2.4.1 Commutation Matrices, Kronecker Products, and Vecs 2.4.2 Commutation Matrices and Cross-Products2.5 Generalized Vecs and Rvecs of the Commutation Matrix; 2.5.1 Deriving Results for Generalized Vecs and Rvecs of the Commutation Matrix; 2.5.2 Generalized Vecs and Rvecs of the Commutation Matrix and Cross-Products; 2.5.3; 2.5.4 The Matrix; 2.6 The Matrix; 2.7 Twining Matrices; 2.7.1 Introduction; 2.7.2 Definition and Explicit Expressions for a Twining Matrix; 2.7.3 Twining Matrix and the Commutation Matrix; 2.7.4 Properties of the Twining Matrix .; 2.7.5 Some Special Cases; 2.7.6 Kronecker Products and Twining Matrices; 2.7.7 Generalizations A More General Definition of a Twining Matrix2.7.8 Intertwining Columns of Matrices; Three Elimination and Duplication Matrices; 3.1 Introduction; 3.2 Elimination Matrices; 3.2.1 The Elimination Matrix; 3.2.2 The Elimination Matrix; 3.2.3 The Elimination Matrices and; 3.2.4 The Elimination Matrices; 3.3 Duplication Matrices; 3.3.1 The Duplication Matrix; 3.3.2 The Elimination Matrix and the Duplication Matrix; 3.3.3 The Duplication Matrix; Four Matrix Calculus; 4.1 Introduction; 4.2 Different Concepts of a Derivative of a Matrix with Respect to Another Matrix 4.3 The Commutation Matrix and the Concepts of Matrix Derivatives4.4 Relationships Between the Different Concepts; 4.5 Tranformation Principles Between the Concepts; 4.5.1 Concept 1 and Concept 2; 4.5.2 Concept 1 and Concept 3; 4.5.3 Concept 2 and Concept 3; 4.6 Transformation Principle One; 4.7 Transformation Principle Two; 4.8 Recursive Derivatives; Five New Matrix Calculus Results; 5.1 Introduction; 5.2 Concept of a Matrix Derivative Used; 5.3 Some Basic Rules of Matrix Calculus; 5.4 Matrix Calculus Results Involving Generalized Rvecs or Cross-Products 5.5 Matrix Derivatives of Generalized Vecs and Rvecs5.5.1 Introduction; 5.5.2 Large X; Results for Generalized rvecs; Results for Generalized vecs; 5.5.3 Small X; Results for Generalized rvecs; Result for Generalized vecs; 5.6 Matrix Derivatives of Cross-Products; 5.6.1 Basic Cross-Products; 5.6.2 Cross-Products Involving; 5.6.3 Cross-Products Involving; 5.6.4 The Cross-Product; 5.6.5 The Cross-Product; 5.6.6 The Cross-Product; 5.7 Results with Reference to; 5.7.1 Introduction; 5.7.2 Simple Theorems Involving; 5.7.3 Theorems Concerning Derivatives Involving VecA, VechA, and 5.7.4 Theorems Concerning Derivatives Involving VecX where X Is Symmetric Literaturverz. S. 257 - 258
ISBN:	978-1-139-42440-0 ; 1-139-42440-8 ; 978-1-107-03200-2 ; 978-1-107-44872-8
Other identifiers:	10.1017/CBO9781139424400 [DOI]
Classification:	Lineare Algebra, multilineare Algebra
Source:	ECONIS - Online Catalogue of the ZBW

Persistent link: https://www.econbiz.de/10015621519