By Muir T.

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2 . 1 3 a ) and ( 2 . e. 12) The relationships between the Jordan chains of A C and Jordan cha£ns of Dr(X) can be d e s c r i b e d in the f o l l o w i n g i m p o r t a n t theorems. 5 a latent If qij' r o o t Xi, 0£J~£i-1' then q c i j ' i s a r i g h t Jordan c h a i n of Dr(k) a s s o c i a t e d with 0£J££i-I' a right Jordan c h a i n of Ac a s s o c i a t e d with an eigenvalue li, can be found as = qclj ! 1 $(k)(ki) q . . . 13a) where ~k)(k)~d(k)~r(k)/dkk " n - t h d e r i v a t e of ~ r ( k ) w i t h r e s p e c t to k and Sr(X) i s d e f i n e d i n Eq.

3 , 3 . 11 Let D(~) be a n o n s l n g u l a r k - m a t r l x and DCI) " D (1)U ( 1 ) r r where Dr(k) i s a column-reduced c a n o n i c a l k - m a t r i x and Ur(1) i s un~modular. Let i i be a l a t e n t and r o o t of Dr(1) , P i j and q l j for 0

1) is an observable system. D~(k) and N~(k) are Note t~at avy row of N%(k) Co Vi=0 is a zero row. Prom Eqs. 14), the minimal realization of D-I( ~,,, using a pair r (A,B) can be formulated as follows. e. V l(k) ffi Cr(kln-A)-iB+Dr. Proof: Prom Eqs. 23a) 25 Also, from Eq. ( 2 . =0 1 Thus~ m ~i=O Since DrCX) = Drh6(k) , we have D-I(X)r = ~ - I ( A ) D ~ " we have When < i - 0 , from Eq. 14e) {6(X)}i i = I, {6(X)}i j = 0 for i~j, and {6(A)}q i = 0 for q~i. As a result, when *
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