Derivatives of the characteristic root of a synmetric or a hermitian matrix with two applications in multivariate analysis

Derivatives of the α th largest sharactsristic root of a symmetric matrix S = (srs) with respect to srs (r ≦ s) at S = A = diag(λ1,…, λp) are given in this paper, whers λ1 ≧… ≧ λp and λα is assumed to be simple.The first application lies in deriving the partial differential equation for zonal polynomials given by James [13] and further new partial differential aquation of fourth degree for zonal polynomials.The second application lies inegiving the asymptotic expansions of the distribution of the a, th largest root of a Nishart matrix having Wp (n, ∑), when a th root of ∑ is simple.It is given by normal distribution function and its derivatives, If the a th root is not simple, non-normal limitiing distribution is obtained when p = 2, The similar results for the derivatives of a Heraitianmatrix and for a root or a complex Wishart matrix are also given.