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Symmetry, 0-1 Matrices and Jacobians: A Review

Published online by Cambridge University Press:  18 October 2010

Jan R. Magnus  and
H. Neudecker
Jan R. Magnus
Affiliation:
London School of Economics
H. Neudecker
Affiliation:
University of Amsterdam
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Abstract

In this paper we bring together those properties of the Kronecker product, the vec operator, and 0-1 matrices which in our view are of interest to researchers and students in econometrics and statistics. The treatment of Kronecker products and the vec operator is fairly exhaustive; the treatment of 0–1 matrices is selective. In particular we study the “commutation” matrix K (defined implicitly by K vec A = vec A′ for any matrix A of the appropriate order), the idempotent matrix N = ½ (I + K), which plays a central role in normal distribution theory, and the “duplication” matrix D, which arises in the context of symmetry. We present an easy and elegant way (via differentials) to evaluate Jacobian matrices (first derivatives), Hessian matrices (second derivatives), and Jacobian determinants, even if symmetric matrix arguments are involved. Finally we deal with the computation of information matrices in situations where positive definite matrices are arguments of the likelihood function.

Type
Articles
Information
Econometric Theory , Volume 2 , Issue 2 , August 1986 , pp. 157 - 190
Copyright
Copyright © Cambridge University Press 1986

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