Symmetry, 0-1 Matrices and Jacobians: A Review

Jan Magnus () and H. Neudecker

Econometric Theory, 1986, vol. 2, issue 2, 157-190

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.

Date: 1986
References: Add references at CitEc
Citations: View citations in EconPapers (27)

Downloads: (external link)
https://www.cambridge.org/core/product/identifier/ ... type/journal_article link to article abstract page (text/html)

Related works:
Working Paper: SYMMETRY, 0-1 MATRICES, AND JACOBIANS: A REVIEW (1985)
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:cup:etheor:v:2:y:1986:i:02:p:157-190_01

Access Statistics for this article

More articles in Econometric Theory from Cambridge University Press Cambridge University Press, UPH, Shaftesbury Road, Cambridge CB2 8BS UK.
Bibliographic data for series maintained by Kirk Stebbing ().