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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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References

1. Aitken, A. C.On the Wishart distribution in statistics. Biometrika 36 (1949): 59–62.CrossRefGoogle ScholarPubMed
2. Balestra, P.La derivation matricielle. Collection de l'lnstitut de Mathématiques Economiques, No. 12. Paris: Sirey, 1976.Google Scholar
3. Barnett, S.Matrix differential equations and Kronecker products. SIAM Journal of Applied Mathematics 24 (1973): 1–5.Google Scholar
4. Bellman, R.Introduction to matrix analysis, 2nd edition. New York: McGraw-Hill, 1970.Google Scholar
5. Browne, M. W.Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal 8 (1974): 1–24.Google Scholar
6. Deemer, W. L. & Olkin, I.. The Jacobians of certain matrix transformations useful in multivariate analysis. Biometrika 38 (1951): 345–367.Google Scholar
7. Don, F.J.H. & Plas, A. P. van der. A note on the characteristic polynomial of the commutation matrix. Linear Algebra and Its Applications 37 (1981): 135–142.Google Scholar
8. Hartwig, R. E. & Morris, S. B.. The universal flip matrix and the generalized faro-shuffle. Pacific Journal of Mathematics 58 (1975): 445–455.Google Scholar
9. Henderson, H. V. & Searle, S. R.. Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics. The Canadian Journal of Statistics 7 (1979): 65–81.Google Scholar
10. Henderson, H. V. & Searle, S. R.. The vec-peraiutation matrix, the vec operator and Kronecker products: a review. Linear and Multilinear Algebra 9 (1981): 271–288.CrossRefGoogle Scholar
11. Hillier, G. H.On the joint and marginal densities of instrumental variable estimators in a general structural equation. Econometric Theory 1 (1985): 53–72.CrossRefGoogle Scholar
12. Jack, H.Jacobians of transformations involving orthogonal matrices. Proceedings of the Royal Society of Edinburgh (Section A) 67 (1966): 81–103.Google Scholar
13. Koopmans, T.C.Rubin, H., & Leipnik, R. B.. Measuring the equation systems of dynamic economics. In Koopmans, T. C. (ed.), Statistical inference in dynamic economic models. New York: John Wiley, 1950.Google Scholar
14. Ledermann, W.On singular pencils of Zehfuss, compound, and Schlaflian matrices. Proceedings of the Royal Society of Edinburgh 56 (1936): 50–89.CrossRefGoogle Scholar
15. MacDuffee, C. C.The theory of matrices. Berlin, 1933, Reprinted by Chelsea, New York.CrossRefGoogle Scholar
16. Magnus, J. R.L-structured matrices and linear matrix equations. Linear and Multilinear Algebra 14 (1983): 67–88.CrossRefGoogle Scholar
17. Magnus, J. R. & Neudecker, H.. The commutation matrix: some properties and applications. The Annals of Statistics 1 (1979): 381–394.Google Scholar
18. Magnus, J. R. & Neudecker, H.. The elimination matrix: some lemmas and applications. SIAM Journal on Algebraic and Discrete Methods 1 (1980): 422–449.CrossRefGoogle Scholar
19. Magnus, J. R. & Neudecker, H.. Matrix differential calculus with applications to simple, Hadamard, and Kronecker products. Journal of Mathematical Psychology 29 (1985): 474–492.CrossRefGoogle Scholar
20. Muirhead, R. J.Aspects of multivariate statistical theory. New York: John Wiley, 1982.Google Scholar
21. Murnaghan, F. D.The theory of group representations. Baltimore: Johns Hopkins Press, 1938.Google Scholar
22. Nel, D. G.On the symmetric multivariate normal distribution and the asymptotic expansion of a Wishart matrix. South African Statistical Journal 12 (1978): 145'159.Google Scholar
23. Neudecker, H.The Kronecker matrix product and some of its applications in econometrics. Statistica Neerlandica 22 (1968): 69–81.Google Scholar
24. Neudecker, H.Some theorems on matrix differentiation with special reference to Kronecker matrix products. Journal of the American Statistical Association 64 (1969): 953–963.CrossRefGoogle Scholar
25. Neudecker, H.On Jacobians of transformations with skew-symmetric, strictly (lower) triangular or diagonal matrix arguments. Linear and Multilinear Algebra 14 (1983): 271–295.Google Scholar
26. Neudecker, H. & Wansbeek, T.. Some results on commutation matrices, with statistical applications. The Canadian Journal of Statistics 11 (1983): 221–231.CrossRefGoogle Scholar
27. Olkin, I.Note on ‘The Jacobians of certain matrix transformations useful in multivariate analysis’. Biometrika 40 (1953): 43–46.Google Scholar
28. Olkin, I. & Sampson, A. P.. Jacobians of matrix transformations and induced functional equations. Linear Algebra and Its Applications 5 (1972): 257–276.CrossRefGoogle Scholar
29. Phillips, P.C.B.The exact distribution of LIML: I. International Economic Review 25 (1984): 249–261.CrossRefGoogle Scholar
30. Phillips, P.C.B.The exact distribution of LIML: II. International Economic Review 26 (1985): 21–36.Google Scholar
31. Pollock, D.S.G.The algebra of econometrics. New York: John Wiley, 1979.Google Scholar
32. Pollock, D.S.G.Tensor products and matrix differential calculus. Linear Algebra and Its Applications 67 (1985): 169–193.Google Scholar
33. Richard, J. F.A note on the information matrix of the multivariate normal distribution. Journal of Econometrics 3 (1975): 57–60.CrossRefGoogle Scholar
34. Rogers, G. S.Matrix derivatives. New York: Marcel Dekker, 1980.Google Scholar
35. Roth, W. E.On direct product matrices. Bulletin of the American Mathematical Society 40 (1934): 461–468.CrossRefGoogle Scholar
36. Sylvester, J.Sur la solution du cas le plus général des équations linéaires en quantités binaires, c'est-à-dire en quaternions ou en matrices du second ordre. Comptes Rendus des Séances de l' Académic des Sciences 99 (1884): 117–118.Google Scholar
37. Sylvester, J.Sur la resolution générale de l'équation linéaire en matrices d'un ordre quelconque. Comptes Rendus des Séances de l'Académic des Sciences 99 (1884): 409–412 and 432-436.Google Scholar
38. Tracy, D. S. & Dwyer, P. S.. Multivariate maxima and minima with matrix derivatives. Journal of the Americal Statistical Association 64 (1969): 1576–1594.Google Scholar
39. Tracy, D. S. & Singh, R. P.. Some modifications of matrix differentiation for evaluating Jacobians of symmetric matrix transformations. In Tracy, D.S. (ed.), Symmetric functions in statistics. University of Windsor, Ontario, Canada, 1972.Google Scholar
40. Vartak, N. N.On an application of Kronecker product of matrices to statistical designs. Annals of Mathematical Statistics 26 (1955): 420–438.Google Scholar
41. Vetter, W. J.Vector structures and solutions of linear matrix equations. Linear Algebra and Its Applications 10 (1975): 181–188.Google Scholar
42. Wiens, D. P.On some pattern-reduction matrices which appear in statistics. Linear Algebra and Its Applications 67 (1985): 233–258.CrossRefGoogle Scholar
43. Zehfuss, G.Ueber eine gewisse determinante. Zeitschriftfiir Mathematik und Physik 3 (1858): 298–301.Google Scholar