596 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



6.02 We observe at once that if a 6(L) exists satisfying (i) and (ii), then 

 it must be unicjue because it is exactly the minimum number of reactive 

 elements required to realize any representative of the Cauer class L. No 

 particular pains then need be taken as we go along to verify that the 

 value of b{L) ai'rived at is in fact independent of the mode of defining 

 it. 



6.1 Given a PR geometrical linear correspondence L between V and K, 

 there is a frame which reduces L in the sense of (I, 13.02). In this frame 

 we have the dual decomposition 



V = Vz.0 e Vo e Vi 



K = Ki Ko e K,.o 



in which each subspace is real and spanned by selected basis vectors. 

 Furthermore, 



K,. = K.. K;.o , 



Finally, if r is the dimension of V> and K2 , there is an r X r PR matrix 

 [Zxii))] such that, when 



[v-i , ko] e L(p) 



and 



V2 eV-i , hi e K2 , 



then 



[v.^ = [Z,{v)]M. 



Here the r-tuples are those representing v-2. and ki as elements of V2 and 

 K2 in the chosen frame. 



6.11 Definition: We define 5(L) by 



6(L) = 5([Zx]), 



where [Zi(p)] is the matrix described above. 



6.12 This number b{L) is the number of reactive elements used when 

 the Brune process is applied to realize [Zi(p)]. (This is 4.07). Then, 

 however, by the argument of (I, 8.5), the representative [L] of L in the 

 particular frame in question can be realized by adjoining open and 

 short circuits to a realization of [Zi(p)]. This operation adds no new 

 reactive elements. Neither does the operation of converting [L] to any 



