Charles Micchelli
Journal of Approximation Theory
Two very basic transformations in multivariate statistics are those of a p×q matrix X to a p×q matrix Y defined by Y=AXB (where A and B are matrices of constants) and of a p×p nonsingular matrix X to a p×p matrix W defined by W=X-1. The Jacobians of these transformations are known to be |A|q|B|p and (-1)p|X|-2p, respectively, or |A|p+1 and (-1)p(p+1)/2|X|-(p+1), respectively, depending on whether X is unrestricted or X is symmetric and B=A′. The derivation of these formulas is greatly facilitated by the introduction of the vec and vech operators [H. Neudecker, J. Amer. Statist. Assoc. 64 (1969) 953-963; H.V. Henderson, S.R. Searle, Canad. J. Statist. 7 (1979) 65-81; J.R. Magnus, H. Neudecker, SIAM J. Algebraic Discrete Methods 1 (1980) 422-449; J.R. Magnus, H. Neudecker, Econometric Theory 2 (1986) 157-190]. Only relatively basic properties of these operators are needed. Arguments that appeal to the existence of the singular value decomposition or to related decompositions are not needed; nor is it necessary to introduce matrix differentials. © 2000 Elsevier Science Inc.
Charles Micchelli
Journal of Approximation Theory
Ehud Altman, Kenneth R. Brown, et al.
PRX Quantum
Nimrod Megiddo
Journal of Symbolic Computation
Daniel J. Costello Jr., Pierre R. Chevillat, et al.
ISIT 1997