On the Subdifferentiability of Functions of a Matrix Spectrum Ii Subdifferenti

James V Burke

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On the other hand, if ^ > Q'*{zo;d), then there is a 7 G r{zo, d) such that a(y{()) - a(zo) urn 0, for every branch A(e) of the splitting of At along the curve 7, and for every k G A(2o). Now by applying Lemma 4 of Part I to this inequality for each k G A{2o) and then letting 6 J. Q'^{zQ\d), we obtain inequality (8). The proof of the last statement requires the implicit function theorem and is given in [4]. N Theorein 2 Choose d £ C and set TZ{zo) := {k : \\k\= p{zo)]. We will consider two cas...es; p(zo) = and p{zo) ^ 0. 1) Let us assume that p{zo) = so that m = 1 and Ai = 0. // any one of the conditions s J2(tT N(-'Bu)d, = 0, j = 2, . . . , Hi, (10) (=1 IS violated, then otherwise, where p''{zo;d) = +00 ; p''{zo;d)>azo, d), (11) azo, d):=-\T{tTBi, )d, \ n '-^ Moreover, if condition (10) holds and the vectors {[trNi-'Bn, ---, UNr'BuV : j=l, ... , n} are linearly independent, then equality holds m (11).
S) Let us assume (hat p{zo) ^ 0.

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