Sec. 5.4] THE GENERAL CHARACTERISTIC-NOISE MATRIX 39 



But, Eq. 5.21 is identical with the definition Eq. 3.15 for N. 



Next, let us relate the general noise matrix Nr of Eq. 5.19 to its partic- 

 ular form in the impedance representation. For this purpose, we note 

 that according to Eq. 5.14 



88^ = Mii^M^ (5.22) 



Then, using Eqs. 5.13, 5.18, and 5.20, we find 



[i! -t]q.-[i| -t]^ 



= m[i I -z]RQr~^R^[l | -z]^M^ 

 = -2M(Z + Z+)Mt (5.23) 



Combining Eqs. 5.21, 5.22, and 5.23 with 5.19, we have finally 



Nr = M'^-^NzM'^ (5.24) 



According to Eq. 5.24, the characteristic-noise matrix Ny of the general 

 matrix representation of a network is related by a similarity transforma- 

 tion to the characteristic-noise matrix N^ of the impedance-matrix repre- 

 sentation of the same network. Therefore, Ny and N^ have the same 

 eigenvalues. 



The eigenvalues of the characteristic-noise matrix of Eq. 5.21 deter- 

 mined the stationary values of the real quantity p in Eq. 3.10. Com- 

 parison of these two expressions with the expression for the general 

 characteristic-noise matrix in Eq. 5.19 shows that its eigenvalues deter- 

 mine the stationary values of the associated real quantity pT 



yt[l , -t]Qj^-i[i ; -T] y 



with respect to variations of the arbitrary column matrix y. It is easily 

 shown by the method of Sec. 3.2 that this "noise parameter" pr has, in 

 fact, as extrema the eigenvalues of the matrix Nr defined in Eq. 5.19. 

 The range of values that pr assumes as a function of y is identical with 

 the range of values of p in Eq. 3.10 and Fig. 3.1. 



The network classification in the three cases illustrated in Fig. 3.1 

 should now be restated in terms of the T-matrix representation. This is 

 easily accomplished by considering Eq. 5.23. According to it, the matrices 



[l I -t] Qt~^ [l 1 -t]^ and -(Z 4- Z^) are related by a colinear 



transformation, A colinear transformation preserves the signature of 

 a matrix. Consequently, the network classification of Table 3.1 can be 

 carried out in the T-matrix representation, as shown in Table 5.1. The 

 same conclusion may be reached from the facts that Nr and N^ have the 

 same eigenvalues, and both EE^ and 85^ are positive definite. 



