FORMAL REALIZABILITY THEORY — II 543 



That is, if the impedance matrix of a 2n-pole N is non-singular, then its 

 admit tancp matrix exists, and conversely. 



2.08 It was provcnl by C'auer'\ and in (I, 16.8), that if Z(p) is PR 

 and of rank m < n, then there exists a real, constant, non-singular 

 matrix W such that 



Zip) = W'Z"{p)W (1) 



where Z"(p) is a non-singular )n X m PR matrix bordered by zeros. 



2.09 Properties (i) through (i\') of 1.1 define the PR property for a 

 matrix Z(p). A convenient equivalent definition is that 



(i) Z(p) is symmetric, 

 (ii) For each A: e K, the function 



<p(p) =^ (Z(p)k, k) 



is a PR function of p. 



This equivalent definition was established in (I, 16.13). 



2.1 In Section 16 of Part I it was also mentioned that there exists for 

 any rational operator Z{p) (PR or not) a numerical function 8(Z) 

 which generalizes to operators the usual definition of the degree of a 

 rational function. We list here the properties of this degree 8(Z). They 

 will be established in Section 7. 



2.11 d(Z) is an integer > 0. 



2.12 If 5(Z) = 0, then Z(p) is a constant — that is, does not depend 

 upon p. 



2.13 If Z~\p) exists, then 5(Z) = 5(Z~'). 



2.14 If Z(p) = Zi(p) -f Z2(p), where Zi{p) is finite at every pole of 

 Z-iip), and Z2(p) is finite at every pole of Zi(p), then 



5(Z) = 5(Zi) -f 5(Z2). 



2.15 If Z(p) = f(p)R, where /(p) is a scalar and .R is a constant operator, 

 then 



5(Z) = [degree of/] -[rank of 72]. 

 Here the degree of / is the sum 



y] [order of the pole of f(p) at po] 



Pa 



where po runs over all poles oi j{p), including oo. 



