548 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



III. A SIMPLE REALIZ ABILITY THEOREM 



3.0 The follo\Aing theorem is contained in Cauer^ Since it provides the 

 basic step in our reaUzability process, we shall prove it here. 



3.1 Theorem: Let f(p) be any one of the four functions 



(i) fip) - 1, 

 (ii) fip) = p, 



Let 72 be a real, constant, symmetric semidefinite n X n matrix of 

 rank r. Then: 



(A) The matrix 



Zip) = fip)R 



is PR and there exists a finite passive 2/i-pole N with the impedance 

 matrix Zip). 



(B) The 2n-pole N can be realized with ideal transformers and, re- 

 spectively, 



(i) with r resistors, 



(ii) with r coils, 



(iii) with r capacitors, 



(iv) with r coils and r capacitors. 



(C) The dual statements to (A) and (B) are true. 



Proof: That Zip) in PR is easily verified directly. It will follow also 

 from the results of Part I when we exhibit a (finite passi^•e) network 

 whose matrix is Zip). To construct this latter, let Z) be a diagonal matrix 

 such that 



R = WDW 



where W is a real, constant, non-singular matrix. That Z) and W always 

 exist is the analog for impedance operators of the result of Hahnos , par. 

 41, for dimensionless operators. In fact, W can be taken to be orthogonal 

 (Tr~^ = TF', cf. Halmos'*, par. 63). If R is of rank r, T> has r non-vanishing 

 diagonal elements, say rfn , c?22 , • • • , d„ . 



Since R is semidefinite, each dafip), 1 < i < r, is the impedance of 

 an ob\'iously passive two pole. Call this two-pole M, . Let Mr+i , • • • , 

 Mn be two poles consisting of short cu'cuits. Consider the 2n-pole Ni 



