H. Neudecker

The Elimination Matrix: Some Lemmas and Applications

Two transformation matrices are introduced, L and D, which contain zero and unit elements only. If A is an arbitrary

Matrix differential calculus with applications to simple, Hadamard, and Kronecker products

( n,n )

Symmetry, 0-1 Matrices and Jacobians: A Review

matrix, L eliminates from vecA the supradiagonal elements of A, while D performs the inverse transformation for symmetricA. Many properties of L and D are derived, in particular in relation to Kronecker products. The usefulness of the two matrices is demonstrated in three areas of mathematical statistics and matrix algebra: maximum likelihood estimation of the multivariate normal distribution, the evaluation of Jacobians of transformations with symmetric or lower triangular matrix arguments, and the solution of matrix equations.

On the identification of restricted factor loading matrices: an alternative condition

Abstract Several definitions are in use for the derivative of an m × p matrix function F(X) with respect to its n × q matrix argument X. We argue that only one of these definitions is a viable one, and that to study smooth maps from the space of n × q matrices to the space of m × p matrices it is often more convenient to study the map from nq-space to mp-space. Also, several procedures exist for a calculus of functions of matrices. It is argued that the procedure based on differentials is superior to other methods of differentiation, and leads inter alia to a satisfactory chain rule for matrix functions.

Matrix Differential Calculus with Applications in Statistics and Econometrics

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.

The Commutation Matrix: Some Properties and Applications

Abstract An alternative sufficient (and under a regularity assumption also necessary) condition for local uniqueness of the solution for restricted factor loading matrices is supplied. It is shown to be equivalent to Bekker, 1986 , Bekker and Pollock, 1986 condition. Although the latter condition is computationally simpler, its derivation is awkward.