24 CHARACTERISTIC-NOISE MATRIX [Ch. 3 



Table 3.1. Classification of Networks and Eigenvalues 



The permissible values of Xs determine the ranges of values that p can 

 assume as a function of x in Eq. 3.10. Let us consider Case 1 first. No 

 eigenvalue is positive. Among the eigenvalues, there is one of least 

 magnitude (possibly zero) and another of largest magnitude. Since ^ is a 



P = -Pe.i 



P = -Pci 



Largest positive 

 eigenvalue "■\. 



"D 



E 



Smallest positive -^ 

 eigenvalue 



-Pe.i = 



Smallest negative 

 eigenvalue 



Intermediate 

 eigenvalues 



Largest negative 

 eigenvalue 



P = -Pe.i 



I Intermediate 

 I eigenvalues 



kTAf 



kTAf 



-Pe.i^O 



:» 

 E * 



Intermediate 

 eigenvalue 



Smallest 



positive 



eigenvalue 



■Pe.i = 



Smallest 



negative 



eigenvalue 



Intermediate 

 eigenvalue 



Z + zt positive definite 

 (a) 



Z + Z''' negative definite 

 (b) 



Z + zt indefinite 

 (c) 



Fig. 3.1. Schematic diagram of permitted values of p for four-terminal-pair networks. 



continuous function of x, its values lie between these two extreme eigen- 

 values, as illustrated in Fig. 3.1a. Analogous reasoning applies to the 

 second case, illustrated in Fig. 3.16. 



Case 3 is a little more involved. The denominator of Eq. 3.10 can 

 certainly become zero for some values of x. Correspondingly, infinite 

 values of p will occur. Among the eigenvalues, there is a smallest 



