Some Theorems on Matrix Differentiation with Special Reference to Kronecker Matrix Products

Abstract

Matrix differentiation, the procedure of finding partial derivatives of the elements of a matrix function with respect to the elements of the argument matrix, is the subject matter of this paper. The method followed here amounts to linking the differentials of the matrix function and the argument matrix, and then identifying matrices of partial derivatives. The basic assumption made is mathematical independence of the elements of the argument matrix. Matrix functions are divided into two categories: Kronecker matrix products and non-Kronecker (= ordinary) matrix products. Three definitions of partial derivatives matrices are used, to be denoted by D ₁, D ₂ and D ₃ in this abstract. D ₂ and D ₃ are applied to Kronecker products, D ₁ is applied to ordinary products. This is a matter of efficiency only. D ₁ is developed first. Transforming matrices into column vectors turns out to be very convenient. D ₂ and D ₃ are developed then. D ₃ is a generalisation of D ₁.