Simple derivations for two Jacobians of basic importance in multivariate statistics

Abstract

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.