Tensor products and matrix differential calculus

Abstract

AbstractThis paper sets forth some of the principal results of the algebra of Kronecker products in a way which relates them directly to the abstract algebra of tensor products. The concepts and the results that are developed in this way are used to analyse three alternative definitions that have been proposed for the derivative of a matrix function Y=Y(X) with respect to its matrix argument X. It is argued that only one of these definitions is viable. The other definitions, which are widely used in econometrics, are not consistent with the classical representation of linear algebra via matrix theory; and they lead to serious practical difficulties that do not arise when the appropriate definition is adopted