Symmetry, 0-1 Matrices and Jacobians: A Review

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.