Generalized Vectorization, Cross-Products, and Matrix CalculusCambridge University Press, 11 feb. 2013 - 267 pagini "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"-- |
Cuprins
ZeroOne Matrices | 28 |
Elimination and Duplication Matrices | 89 |
Matrix Calculus | 134 |
New Matrix Calculus Results | 164 |
Applications | 214 |
Symbols and Operators Used in this Book | 255 |
Alte ediții - Afișează-le pe toate
Generalized Vectorization, Cross-Products, and Matrix Calculus Darrell A. Turkington Previzualizare limitată - 2013 |
Generalized Vectorization, Cross-Products, and Matrix Calculus Darrell A. Turkington Nu există previzualizare disponibilă - 2014 |
Termeni și expresii frecvente
2LnNn 8vec 8vecA 8vecX 8vecX applying Arbitrary far values backward chain rule column vector commutation matrix concepts of matrix covariance matrix cross-products defined definition duplication matrices DY(X econometric elementary matrices elements elimination matrix Equation 2.8 following theorem given by Equation Hessian matrix identity matrix intertwining involving iterative procedure jth column Knnn Kronecker products LetA likelihood function LnNn matrix calculus results matrix derivative matrix given matrix of constants maximum likelihood estimator notation nuisance parameters null vector obtain partial derivatives partitioned matrices permutation matrix premultiplying Proof properties pX q matrix rvec operators rvecA rvecnKGn scalar function selection matrices side of Equation stacking submatrix Suppose symmetric matrix test statistic Theorem Theorem 2.3 Transformation Principle Turkington 2005 twining matrix vec(A vecA vech vechA vecs and rvecs vecX write X n matrix Ywith respect zero-one matrices