FORMAL REALIZABILITY THEORY — I 239 



8.5* Tlie sufficiency half of 8.0 is now clear. By 8.2 and 8.8, Ni is physi- 

 call.y realizable. Clearly then N w is, simply by the adjunction of the 

 necessary open and short circuits. Finally N is by Cauer's theorem, 8.4. 



8.0* We can see now how to prove the necessity of positive reality for 

 the realizability of a positive real matrix Z(p). This is the necessit}^ half 

 of the matrix theorem 1.1. Let Z(p) be the matrix of a realizable N. 

 Then N has an associated linear correspondence L satisfying PI, • • • , P7, 

 by tiie necessity half of 8.0. The pairs of L are the pairs 



[Z{p)k, k] 



generated as A; ranges over all /^-tuples. By definition, then, the pairs of 

 L w are 



[W~'Z{p)k, W'k]. 



As k ranges over all n-tuples, the nonsingularity of W implies that 

 W'k does also. Let U = W . Then the pairs above are the same as 



[UZ{p)U'k,k] 



as k ranges over all n-tuples. Hence Lw has the impedance matrix 

 UZ{p)U', where U = W~ is real and nonsingular. Because Lw has an 

 impedance matrix, r = in 8.12. 



Now by 8.1 and 8.2, Zi{p) is positive real and the matrix UZ{p)U' 

 of L w is just Zi{p) bordered by s rows and columns of zeros. It is then 

 easy to see that UZ{p)U' is positive real, and finally also that Z(p) is. 

 These last two facts will be proved formally in Section 16. 



IX. cauer's transformation theorem 



9.0 Cauer's transformation theorem is the cornerstone of formal reali- 

 zability theory. In one form, the theorem reads: 



9.1* Let Z{p) be the impedance matrix of a physically realizable 2/i-pole 

 N. Let U be a real, constant, nonsingular matrix. Then 



UZ{p)U' (1) 



is again the impedance matrix of a physically realizable 2n-pole, Ny . 

 Nr can be constructed from N by the use of ideal transformers. 



9.2* A superficial generalization of this theorem can be obtained at once 

 from Cauer's proof. It asserts that if N is physically realizable and is 

 described by the linear correspondence L, then there is a physically 

 realizable 2n-pole N »■ , obtainable from N by the use of ideal trans- 



* Technical paragraph as explained in Section 2.91. 



