Sec. 3.3] PROPERTIES OF THE EIGENVALUES 25 



positive one and a negative one of smallest magnitude. (In special cases, 

 one or both may be equal to zero.) Again, because of the continuous 

 nature of /? as a function of x, p may never take a value between and dis- 

 tinct from the foregoing extreme eigenvalues. The gap between the 

 ranges of allowed values of p is illustrated in Fig. 3.1c. 



One particular property of the eigenvalues of N will be of importance 

 later. Suppose that the original network with the characteristic-noise 

 matrix N is imbedded in a 2w-terminal-pair lossless network, as shown in 

 Fig. 2.2. A new w-terminal-pair network results, with the characteristic- 

 noise matrix N'. The eigenvalues of N' are the stationary values of the 

 exchangeable power obtained in a subsequent imbedding of the type 

 shown in Fig. 2.4. This second imbedding network is completely variable. 

 One possible variation removes the first 2w-terminal-pair imbedding. 

 Accordingly, the stationary values of the exchangeable power at the 

 (w -f l)th terminal pair in Fig. 2.4 do not change when a 2 w- terminal-pair 

 lossless transformation is interposed between the two networks shown. 

 It follows that: 



The eigenvalues of the characteristic noise matrix N are invariant to a 

 lossless transformation that preserves the number of terminal pairs. 



At this point appHcation of our results to two familiar examples of 

 linear networks helps to establish further significance for the character- 

 istic-noise matrix and its eigenvalues. 



If the w-terminal-pair network contains only coherent (signal) gener- 

 ators rather than noise generators, EiEk* = EiEk* because ensemble 

 averaging is unnecessary. The matrix EE"'^ is then of rank one ; that is, a 

 determinant formed of any submatrix of order greater than one is zero 

 because its rows (or columns) are all proportional (Eq. 2.1), The rank 

 of N cannot exceed that of either of its factors, so it too is of rank one 

 (zero in a trivial case). Matrix N therefore has only one nonzero eigen- 

 value, and this is equal to trace N. From Eq. 3.18, we conclude that 

 for such networks, containing only coherent signal generators, the 

 operations defined by Fig. 2.4 lead merely to the exchangeable power for 

 the whole network (in the sense of Eq. 2.15). It is the single stationary 

 value of Pe,i and also the negative of the sole eigenvalue of N. 



Another simple but quite different case arises if the original (non- 

 reciprocal) network is a passive one with dissipation, (Z -|- Z^) positive 

 definite, in thermal equilibrium at absolute temperature T. Then the 

 operations defined by Fig. 2.4 must, on thermodynamic grounds, always 

 lead to Pe,i = kT A/ in a frequency band A/, where k is Boltzmann's 

 constant. No matter what form the variable lossless network may take. 



