Old and New Matrix Algebra Useful for Statistics
The partials with respect to the numerator are laid out according to the shape of Y while the partials with respect to the denominator are laid out according to the transpose of X. For example, dy/dx is a column vector while dy/dx is a row vector (assuming x and y are column vectors—otherwise it is flipped). Each of these derivatives can be tediously computed via partials, but this section shows how they instead can be computed with matrix manipulations. The material is based on Magnus and Neudecker (1988).