E-sufficient statistics : application of Neumann's Lemma to the minimum and characteristic polynomials

Following the theory of sufficient statistics, developed by Halmos and Savage [J], we characterize the surjective e-sufficient statistic for a dominated family of probability measures. This is followed by a characte rization of linear e-sufficient statistic for families of the normal popu lations. Part II. In this paper, first we give a necessary and sufficient condition for which an eigenvector of an n x n matrix is also an eigenvector of its conjugate transpose; later we determine conditions for which the minimum and/or characteristic polynomials of an n x n matrix have simple zeros.