ABSTRACT
Differentiating a vector-valued function of a vector argument with respect to another vector will result in a matrix where the individual elements are the partial derivatives of each element of the function with respect to each element of the vector argument. Most of the basic results can be obtained by expressing the functions of the vectors in terms of the individual elements and expanding, for example, inner products of vectors as sums of products of individual elements. It may help understanding to write out explicitly the cases n = 1 and n = 2. Many of the basic rules of scalar calculus of single variables (e.g., differentiation of products etc.) carry through recognisably to the vector and matrix cases. Symmetry is the most common situation in statistical applications where this arises but skew-symmetric matrices and also matrices of other special structures (diagonal, triangular, banded etc.) need careful handling.
