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

Cambridge: Cambridge University Press, 2013. — 279 p."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"Mathematical prerequisites
Zero-one matrices
Elimination and duplication matrices
Matrix calculus
New matrix calculus results
Applications Preface
one Mathematical PrerequisitesKronecker Products
Cross-Product of Matrices
Vecs, Rvecs, Generalized Vecs, and Rvecs
Basic Operators
Vecs, Rvecs, and the Cross-Product Operator
Related Operators: Vech and
Generalized Vecs and Generalized Rvecs
Generalized Vec Operators and the Cross-Product Operator
two Zero-One MatricesSelection Matrices and Permutation Matrices
The Elementary Matrix
The Commutation Matrix
Commutation Matrices, Kronecker Products, and Vecs
Commutation Matrices and Cross-Products
Generalized Vecs and Rvecs of the Commutation Matrix
Deriving Results for Generalized Vecs and Rvecs of the Commutation Matrix
Generalized Vecs and Rvecs of the Commutation Matrix and Cross-Products
The Matrix
The Matrix
Twining MatricesDefinition and Explicit Expressions for a Twining Matrix
Twining Matrix and the Commutation Matrix
Properties of the Twining Matrix
Some Special Cases
Kronecker Products and Twining Matrices
Generalizations A More General Definition of a Twining Matrix
Intertwining Columns of Matrices
Three Elimination and Duplication MatricesElimination Matrices
The Elimination Matrix
The Elimination Matrix
The Elimination Matrices and
The Elimination Matrices
Duplication Matrices
The Duplication Matrix
The Elimination Matrix and the Duplication Matrix
The Duplication Matrix
Four Matrix CalculusDifferent Concepts of a Derivative of a Matrix with Respect to Another Matrix
The Commutation Matrix and the Concepts of Matrix Derivatives
Relationships Between the Different Concepts
Transformation Principles Between the Concepts
Concept 1 and Concept 2
Concept 1 and Concept 3
Concept 2 and Concept 3
Transformation Principle One
Transformation Principle Two
Recursive Derivatives
Five New Matrix Calculus ResultsConcept of a Matrix Derivative Used
Some Basic Rules of Matrix Calculus
Matrix Calculus Results Involving Generalized Rvecs or Cross-Products
Matrix Derivatives of Generalized Vecs and RvecsLarge X
Results for Generalized rvecs
Results for Generalized vecs
Small X
Results for Generalized rvecs
Result for Generalized vecs
Matrix Derivatives of Cross-Products
Basic Cross-Products
Cross-Products Involving
Cross-Products Involving
The Cross-Product
The Cross-Product
The Cross-Product
Results with Reference toSimple Theorems Involving
Theorems Concerning Derivatives Involving VecA, VechA, and