A Matrix Form of Taylor's Theorem

Abstract

The following pages continue a line of enquiry begun in a work On Differentiating a Matrix, (Proceedings of the Edinburgh Mathematical Society (2) 1 (1927), 111-128), which arose out of the Cayley operator , where xij is the ijth element of a square matrix [xij] of order n, and all n2 elements are taken as independent variables. The present work follows up the implications of Theorem III in the original, which stated thatwhere s (Xr) is the sum of the principal diagonal elements in the matrix Xr. This is now written ΩsXr = rXr – 1 and Ωs is taken as a fundamental operator analogous to ordinary differentiation, but applicable to matrices of any finite order n.