562 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



pected to have an impedance matrix. The following reasoning shows 

 that it does have, however: 



Consider a y e V for which Y''^\p)v = 0. Then from the right side of 

 (9), with (5), 



2v 2 2p- 



V = 2 , 2 Z{p)Gv + ^- — 2 AGv + -r-r—2 BGv. (10) 



p + 0)0 p + Wo p + COo 



We have by hypothesis that Z(p) is finite on p — ico. Therefore w^e may 

 calculate, by letting p -^ in (10), that 



V = -2 AGv, 



COo 



and, by letting p -^ oo in (10), that 



V = 2BGv. 



These two equations exhibit v as an element in the range of A and also 

 an element in the range of B. The only possible such y is y = 0, by 4.43. 

 Therefore there is no non-zero v such that Y^^\p)v = 0. Then Z''^\p) = 

 Y''^\p)~^ exists as a PR operator. 



4.491 Let 



Up) =-H + pF (11) 



P 



be the matrix whose poles at p = and p = 'x, are those of Z^ (p). 

 That is, let 



Z''\p) = L(p) + Z''\p), (12) 



where Z^*^(p) is PR and finite at and oo . Because Z^^\p) is PR, H and F 

 are both real, symmetric, and semidefinite. Let N/, be the 2n-pole whose 

 impedance matrix is L{p), and N^^' the 2/i-pole with matrix Z^(p). 

 In fact, Nl is realizable. N^^^ is the series combination of N^ and N * , 

 by (12). 



4.5 Equations (5), (7), (8), and (12) above are statements about matrices 

 in a particular coordinate frame — that frame appropriate to the given N. 

 We can interpret them as operator relations by simple decree. We wish 

 now to draw a circuit diagram illustrating these relations. To do so, Ave 

 introduce a suitable new coordinate frame. 



Because G(p) is PR and of rank m, we know that a frame can be 

 found in which the matrix for G(p) is an m X m non-singular matrix 

 bordered by zeros (2.08, or (I, 16.8)). By (7) and the result of 4.48, we 



