560 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



Proof: We may assume that the first r diagonal elements of D are the 

 non-zero elements of Z)+ , the next s those of — 7)_ . By (4), 



{W'T'v = mD+Wh , 



{WT'v = - D_Wh . 



The first of these relations exhibits (W')~^v as an n-tuple with non-zero 

 components at most among the first r, the second as an /i-tuple with 

 non-zero components at most among the last ??- — r. Hence all com- 

 ponents of {Wy^v are zero. Hence v itself is zero. 



4.44 Define 



X(p) = -^ A - pB, ' (6) 



V 



and let Nx be the 2w-pole whose impedance matrix is X(p). Nx is not 

 physically realizable, since it is made up of negative reactances. 



Let N^^^ be the 2n-pole whose impedance matrix is Z^'\p). Then by 

 (5) N obtains from N*~^ and Nx by connecting them in series. 



We have the following relation holding on p = ^co, but only thereon 

 since it is only there that X{p) is a pure imaginary: 



7M - - A + coB 



CO 



Z^'^(tco) = i?(ico) + i 



In particular, at two , 



Z^'^tcoo) = R{io:,) + r[/(rcoo) - W'D+W -|- W'DJ[V] 



= R{icoo), 



by (3) and (4). Since J is the null space of R(i<j:o) by definition, J is the 

 null space of Z^'\iwo). 



4.45 Now Y^'\p) = [Z^-\p)r^ exists and is PR. Since Z^-\iuo) annihi- 

 lates every element of J, it follows that Y^'^\p) does not exist at p = two — 

 therefore Y^'^\p) has a pole at icco . Hence we may apply AP and repre- 

 sent Y^^\p) as a reactance network, with admittance matrix 



g(p) = :j^^g> (7) 



p + too 

 in parallel with a 2n-pole N^^^ which has an admittance matrLx, say 



Y''\p) = G(p) + Y''\p), (8) 



where Y^^\p) is finite at p = two . 



