258 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



if and only if, as vectors in V and K, 



[v, k]eL{p). 

 13.02 In the real frame of 12.4 let us renumber the basis vectors so that 



Vi, • • ■ ,Vr span Vio , 



Vr+l , • ■ , Vr+m Span V2 , 



Vr+m+i , ' • • ,Vn Span Vi . 

 Then 



h , - • - , kr+m span K2 , 



kr+m+i , • • • , kn Span KlO • 



We say that such a frame reduces L. 



13.1 Let us now interpret the s-th components of [v] and [k] in this frame 

 respectively as the voltage across and the current in an ideal branch 

 j8s of a 2/2-pole N, 1 < s < w. 



By construction, the vectors feV/, in this frame have components 

 ctr+m+i = • • • = fln = 0, sluce Vi , • • • , iv+m spau V/, . At the same time, 

 the components 6r+m+i , • • • , ^n of [A;] may be chosen arbitrarily without 

 altering the fact that [[y], [A']]e[L](p) because of 12.06. Therefore, the 

 ideal branches jSr+m+i , • • • , /3„ can each be realized physically by a short 

 circuit. 



In a dual way, since /Cr+x , • • • , /Cn span K^ , any kiK.^ has components 

 61 , • • • , &r all zero in our chosen frame. Furthermore, the components 

 tti , • • • , ttr of [v\ can be chosen at will. Hence the ideal branches 

 /3i , • • • , jSr can each be realized physically by an open circuit. 



Let Ni now be the 2m-pole whose ideal branches are /3r+i , • • • , ^r+m • 

 Let the pairs [[v], [k]] admitted by Ni at each peV^ be the [[v], [k]], 

 where [v, k]eAI(p) (13.01). The representation just found for N shows 

 that N is physically realizable if and only if Ni is. 



13.11 The matrix [Zi{p)] of 12.54 is the impedance matrix of the 2m- 

 pole Ni . 



13.12 We now show that [Zi{p)] is a positive real matrix. The displayed 

 formulae of 12.41 show (ii) and (iii) of 1.1, and 12.42 shows (i). Now 

 suppose that [v, k]eM{p). Then, as vectors in V and K, [v, k]eL(p) by 

 definition of M{p). Then, however, if k is fixed 



Jk{p) = {v, k) 



