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.