Sec. 3.3] PROPERTIES OF THE EIGENVALUES 23 



3.3. Properties of the Eigenvalues of the Characteristic-Noise 

 Matrix in Impedance Form 



We shall now confirm the assertion made earlier (Sec. 2.3) to the 

 effect that: 



The exchangeable power of the n-terminal-pair network, 



is equal to the algebraic sum of the stationary values of Pe,i, which is alter- 

 natively the negative of the trace of the characteristic-noise matrix N. 



Setting W = f (Z + Z'^)~^, we express the typical ijth. element of the 

 matrix i(Z + Z^y^E^ = WEE^ in the form 



(WEEt),y = Z WaEkEj'' 

 k=i 



so that its trace (sum of diagonal elements) becomes 



Trace (WEE+) = £ WikEkEi"" = -trace (N) (3.16) 



k,l=l 



But Pe of the w-terminal-pair network equals 



Pe = i[E^(Z + Zt)-i]E = E^WE = E Ei*WikEk (3.17) 



k,l=l 



Comparing Eq. 3.17 with Eq. 3.16, we find 



Pe = -trace N = - £ X^ (3.18) 



s 



since the trace of a matrix is the sum of its eigenvalues. 



We must now determine the ranges of values that can be assumed by 

 the eigenvalues Xs of the characteristic-noise matrix N as well as the corre- 

 sponding ranges of Pe,i. We first recall that the eigenvalues Xs determine 

 the stationary values of p in Eq. 3.10. The numerator of this expression 

 is never negative, since A(= EE"*") is positive (semi) definite. Thus, the 

 algebraic sign of p is determined by that of the denominator. This 

 in turn depends upon the definite character of B, which is equal to 

 — 2(Z -|- Z^). As noted previously, three cases have to be distinguished, 

 in accordance with the second column of Table 3.1. In the first case, the 

 denominator is always negative. Accordingly, the eigenvalues Xs pertain- 

 ing to this case must all be negative, as shown in the last column of the 

 table. The other cases follow in a similar manner. 



