Sec. 3.2] 



EIGENVALUE FORMULATION 



21 



The significant point now is that t is actually any square matrix of 

 order n because Z^a in Eq. 3.4 is entirely unrestricted! Therefore, a new 

 column matrix x may be defined as 



= T| = 



Xi 



(3.8) 



in which the elements take on all possible complex values as the lossless 

 transformation network Zr in Fig. 2.2 is varied through all its allowed 

 forms. Consequently, the stationary values of Pe,i in Eq. 3.7 may be 

 found most conveniently by determining instead the stationary values 

 of the (real) expression 



^^''"2xt(Z + Z+)x ^^-^^ 



as the complex column matrix x is varied quite arbitrarily. 



Aside from the uninteresting possibility of a lossless original network 

 Z, three cases must be distinguished in Eq. 3.9, corresponding to the 

 three different characters of Z discussed previously in connection with 

 Eqs. 2.13 and 2.15. Since EE^ is positive (semi) definite, these cases are 

 described as follows in terms of the variation of Pe,i as a multidimensional 

 function of all the complex components of x: 



{a) Z + Z^ positive definite; Pe,i > for all x. 

 (&) Z + Z"*^ negative definite; Pe,i < for all x. 

 (c) Z + Z^ indefinite; Pe,i < 0, depending upon x. 



3.2. Eigenvalue Formulation of Stationary-Value Problem 



We now turn to the determination of the extrema and stationary 

 values of Pe,i in Eq. 3.9. For reasons that will become clear in regard to 

 amplifier noise performance, we shall look for extrema of the quantity 

 p = -Pe,i. In terms of A = EE^ and B = -2(Z + Z^), 



P = 



c^EE^ 



cUj 



2x^(Z + Z+)x x^Bx 



(3.10) 



