Symbolic Matrix Derivatives

Let X be the matrix x₌₍, t a scalar, and let X/ t, t/ X denote the matrices x₌₍/ t, t/ x₌₍ respectively. Let Y = y₏ₐ be any matrix product involving X, X' and independent matrices, for example Y = AXBX'C. Consider the matrix derivatives Y/ x₌₍, y₏ₐ/ X. Our purpose is to devise a systematic method for calculating these derivatives. Thus if Y = AX, we find that Y/ x₌₍ = AJ₌₍, y₏ₐ/ X = A'K₏ₐ, where J₌₍ is a matrix of the same dimensions as X, with all elements zero except for a unit in the m-th row and n-th column, and K₏ₐ is similarly defined with respect to Y. We consider also the derivatives of sums, differences, powers, the inverse matrix and the function of a function, thus setting up a matrix analogue of elementary differential calculus. This is designed for application to statistics, and gives a concise and suggestive method for treating such topics as multiple regression and canonical correlation.