Differentiability of Inverse Matrices

Abstract

Consider a m x m matrix A consisting of elements aij as independent variables. Let f(A) be a I x I non-singular matrix with its elements as functions of aij. Okamoto (l), Memon and Okamoto (2), and Siotani (3) evaluate the effects of differential operator ê/êaij on some matrices f-I (A) at A = I(m), m x m identity matrix, for purpose of obtaining asymptotic expansions of certain probability distributions. Normally one inverts a matrix before applying differential operator. This paper proposes a method to deal with problems of this nature without inverting matrices. Some illustrations for use in multivariate statistical analysis are given