Symbolic Matrix Derivatives

Abstract

Let $X$ be the matrix $\lbrack x_{mn}\rbrack, t$ a scalar, and let $\partial X/\partial t, \partial t/\partial X$ denote the matrices $\lbrack\partial x_{mn}/\partial t\rbrack, \lbrack\partial t/\partial x_{mn}\rbrack$ respectively. Let $Y = \lbrack y_{pq}\rbrack$ be any matrix product involving $X, X'$ and independent matrices, for example $Y = AXBX'C$. Consider the matrix derivatives $\partial Y/\partial x_{mn}, \partial y_{pq}/\partial X$. Our purpose is to devise a systematic method for calculating these derivatives. Thus if $Y = AX$, we find that $\partial Y/\partial x_{mn} = AJ_{mn}, \partial y_{pq}/\partial X = A'K_{pq}$, where $J_{mn}$ 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 $K_{pq}$ 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.