Symbolic Matrix Derivatives

Let X be the matrix xₘₙ, t a scalar, and let ∂ X/∂ t, ∂ t/∂ X denote the matrices ∂ xₘₙ/∂ t, ∂ t/∂ xₘₙ respectively. Let Y = ypq be any matrix product involving $X, X'$ and independent matrices, for example $Y = AXBX'C$. Consider the matrix derivatives ∂ Y/∂ xₘₙ, ∂ ypq/∂ X. Our purpose is to devise a systematic method for calculating these derivatives. Thus if $Y = AX$, we find that ∂ Y/∂ xₘₙ = AJₘₙ, ∂ ypq/∂ X = A'Kpq, where Jₘₙ 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 Kpq 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.