Generalized vectorization, cross-products, and matrix calculus / Darrell A. Turkington.

Includes bibliographical references and index.

1. Mathematical prerequisites -- 2. Zero-one matrices -- 3. Elimination and duplication matrices -- 4. Matrix calculus -- 5. New matrix calculus results -- 6. Applications.

"This book studies the mathematics behind matrix calculus, and the final chapter looks at applications of matrix calculus in statistics and econometrics"-- Provided by publisher

"In this chapter we consider elements of matrix algebra, knowledge of which is essential for our future work. This body of mathematics centres around the concepts of Kronecker products and vecs of a matrix. From the elements of a matrix and a matrix the Kronecker product forms a new matrix. The vec operator forms a column vector from the elements of a given matrix by stacking its columns one underneath the other. Several new operators considered in this chapter are derived from these basic operators. The operator which I call the cross product operator takes the sum of Kronecker products formed from submatrices of two given matrices. The rvec operator forms a row vector by stacking the rows of a given matrix alongside each other. The generalized vec operator forms a new matrix from a given matrix by stacking a certain number of its columns, taken as a block, under each other, and the generalized rvec operator forms a new matrix by stacking a certain number of rows, again taken as a block, alongside each other. It is well known that Kronecker products and vecs are intimately connected but this connection also holds for rvec and generalized operators as well. The cross sum operator, as far as I know, is being introduced by this book. As such, I will present several theorems designed to investigate the properties of this operator. The approach I have taken in this book is to list, without proof, well-known properties of the mathematical operator or concept in hand. If, however, I am presenting the properties of a new operator or concept, if I am presenting a property in a different light, or finally if I have something new to say about the concept, then I will give a proof"-- Provided by publisher

Print version record.

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 Transformation 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.

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