Translator Disclaimer
December, 1948 Symbolic Matrix Derivatives
Paul S. Dwyer, M. S. Macphail
Ann. Math. Statist. 19(4): 517-534 (December, 1948). DOI: 10.1214/aoms/1177730148

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.

Citation

Download Citation

Paul S. Dwyer. M. S. Macphail. "Symbolic Matrix Derivatives." Ann. Math. Statist. 19 (4) 517 - 534, December, 1948. https://doi.org/10.1214/aoms/1177730148

Information

Published: December, 1948
First available in Project Euclid: 28 April 2007

zbMATH: 0032.00103
MathSciNet: MR27254
Digital Object Identifier: 10.1214/aoms/1177730148

Rights: Copyright © 1948 Institute of Mathematical Statistics

JOURNAL ARTICLE
18 PAGES


SHARE
Vol.19 • No. 4 • December, 1948
Back to Top