ISRAEL JOURNAL OF MATHEMATICS, Vol. 35, Nos. 1-2, 1980 ON POINTWISE AND ANALYTIC SIMILARITY OF MATRICES BY SHMUEL FRIEDLAND* ABSTRACT Let A(e) and B(e) be complex valued matrices analytic in e at the origin. A(e)-pB(e) if A(e) is similar to B(e) for any le[<r, A(e)~,B(e) if B(e) = T(e)A(e)T-l(e) and T(e) is analytic and IT(e)l # 0 for le I <: r! In this paper we find a necessary and sufficient conditions on A(e) and B(e) such that A(e)~,B(e) provided that A(e)-~B(e). This problem arises in study of certain ordinary differential equations singular with respect to a parameter e in the origin and was first stated by Wasow. 1. Introduction Let A(e) and B(e) be n • n complex valued matrices analytic in a parameter e, in D, = {z, I z [ < r} for some r > O. We call such matrices analytic at the origin. That is we have the Mclaurin expansions (1.1) A(e)=~-'~AkE k, B(~l=~Bke k, Ak, BkeM.(C) k =0 k =0 which converge in D,. One says that A (e) and B (e) are pointwise similar in D, (denote it by A(e)-pB(e)) if A(e) and B(e) are similar for any e E Dr. A(e) and B(e) are said to be analytically similar in Dr, (denote it by A(e) -aB(e)) if there exists T(e), (1.2) T(e) = ~ Tke k, Tk E M. (C) (convergent for I e I < r'), k=O such that (1.3) I T(e)l # 0 for Is 1< r' (here by I TI we denote the determinant of T) and , Sponsored by the United States Army under Contract No. DAAG29-75-C--0024. Received March 16, 1979 89