260 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



ates from K to V, an admittance dually. The operators in Halmos^ 

 are physically dimensionless, in that they operate, e.g., from V to V. 

 This difference is scarcely noticeable. 



We shall regularly omit the duals to concepts or proofs given in terms 

 of impedances. In doing so, we adopt the rule that the dual to an 

 expression 



(Zk, k) 

 is 



(v, Yv). 



14.1 An operator is called symmetric if Z = Z'. Such operators have 

 three useful special properties : 



14.11 If Z is symmetric and ^" and A; are real, then 



(Zj, k) = {Zj, k) ^ iiZrkJ) = (Z'kJ) = iZkJ) 

 by (10) of 10.33, 14.02, 14.01, 14.03, and hypothesis. 



14.12 Let k = ki-\- ik^ , where ki and A:2 are real (10.42). If Z is symmetric 

 then 



(Zk, k) = (Zk, , k,) + iZk2 , k,), 



for, by 14.11, 



(ZA-i , ih) = -i{Zki , k.) = -i(Zk2 , ki) 



= -{Z(ik,),h). 

 (Cf. the similar identity in 12.11.) 



14.13 The symmetric operator Z is completely defined by the quadratic 

 form 



{Zk, k) (1) 



as a function of real keK. For 14.11 permits the formula (29) of 12.42 

 in any real frame, where Us = Zks . The matrix elements of [Z{p)] in 

 that frame are then defined by that formula in terms of values of(l) 

 for real k. 



The form (1) specifies any Z (symmetric or not) if k is allowed to 

 range over all of K (Halmos , par. 53). 



14.2 Let Z(p) now be an impedance operator depending on p. We say 

 that po 9^ <x) is a pole of order m of Z(p) if 



^(A;) = lim ip - p,)"\Z{p)k, k) (2) 



p-*Po 



