554 THE BELL SYSTEM TECHXICAL JOURNAL, MAY 1952 



for Ni , since at least Zi(p) exists. Furthermore, by construction Zi(p) 

 is again an n X n matrix, so rii = n. 

 By 2.14 and 2.15, 



K 



8{Z) = rank {RJ + rank (7^0) + 2 E rank (R,) + 5(Zi). (2) 



k=l 



Since 5(Z) is finite, this shows that K is finite. Indeed, 2K < 8{Z). 

 Furthermore, 8{Z) > 8{Zi), because a matrix of rank zero is itself zero, 

 and by hypothesis Z(p) has a pole on p = ioi. Therefore we have estab- 

 lished the claims in the first line of the Table 4.04, and by a dual argu- 

 ment those in the second line. 



We must yet show that if Ni is physically realizable, then N is. Each 

 term in (1), save Zi(p), is the matrix of a physically realizable 2n-pole, 

 by 3.1. There are at most K -{- 2 such terms. The series combination of 

 their respective 2n-poles is therefore physically realizable and N results 

 from the series connection of these and Ni (2.2). Therefore if Ni is real- 

 izable, so is N. 



Fig. 3 shows the relation of N and Ni under IP, and the dual rela- 

 tion under AP. Here we have shown n = 3. The boxes labelled 0, », 

 • • , K are the devices corresponding to the poles at 0, °° , • • • , *cojc , 

 the terminals on the extreme left are those of N, and Ni is on the right. 



4.11 From (2), and (B) of 3.1, we see that the number of reactive 

 elements used in the realization of the network between Ni and N is 

 exactly 



5(Z) - 5(Zi) = 5(N) - 5(Nx). 



Clearly the dual result holds for AP. This verifies 4.07 for IP and AP. 



4.2 Consider a 2n-pole N in (1 -|- 2, 0). In particular, then, the imped- 

 ance matrix Z{p) of N exists and is not constant, but Z{p) has no in- 

 verse. Then 2.08 applies, and we have 



Z(p) = W'Z!{p)W, (1) 



where W is real, constant, and non-singular, and Zi (p) is a non-singular 

 matrLx Zi(p) bordered by zeros. Let Ni be the 2ni-pole whose impedance 

 matrix is Zi(p). We define ID as the operation which gives Ni from N. 

 Now Hi < n, because Z{p) is singular and Zi(p) is not. Also, Zi(p) is 

 not constant, because Z{p) is not, and 5(Zi) = 5(Z), by 2.17. Therefore 

 ni 9^ 0, also Ni is not in C. Because Zi(p)"^ exists, Ni is in A = 1,2 or 

 3. Because Z(p) has no poles on p = z'w, neither has Zi(p), so Ni e (1 -f- 



