Generalized Vectorization, Cross-Products, and Matrix CalculusCambridge University Press, 11 feb. 2013 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. |
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