164 BELL SYSTEM TECHNICAL JOURNAL 



where now/(X) represents the matrix 



/(X) = Z12X2 - (Zn + Z22)X + Z21 (2.54) 



Let qr be proportional to any non-zero column in F(Xr) where F(\) is the 

 adjoint of /(X) and let g^ be proportional to any non-zero row of -F(Xr). 

 The matrices Q and Q in the expressions (2.45) for v{n) and i{n) are given by 



Q= [gi,?2, ••• gm] 



(2.55) 



where g^ is the column obtained by transposing the row g^ • From (2.45) 

 and (2.53) equations for P and P in terms of Q and Q are obtained: 



P = ZuQ - ZnQA = -Z22Q + Z2,QA~' 



(2.56) 

 P = -ZnQ + Za2QA~' = Z22Q - Z21QA 



The difference equations (2.53) are satisfied by our expressions for v(n) and 

 i(n) by virtue of the relations 



ZuQA' - (Zn -]- Z22)()A -f Z21Q = 



(2.57) 

 Z12QA"' - (Zn + Z22)QA ' + ZoiQ = 



which are a consequence of the properties of the individual columns of 

 Q and Q. 



If the system extends to w = + cc and if the voltages and currents are to 

 remain finite at w = 00 the elements of a must be zero and the expressions 

 (2.45) for v{n) and i(n) become 



v(n) = PA"c = PK!'p-\{o) 



i{n) = QA^a = QA^^Q'^io) (2.58) 



v(n) = PQ~%i), i{n) = QP~\in) 



where we have assumed that P^ and Q~^ exist. We accordingly introduce 

 the characteristic impedance and admittance matrices Zo and Y„ associated 

 with propagation in the positive direction, i.e., in the direction of n in- 

 creasing. 



v{n) = Zoi{n), i{n) = iXw), Z<, = IT 

 Z, = PQ"' = Zn - ZuQAQ-' = -Z22 + Z,,QA-'Qr' (2.59) 

 Yo = QP~' = I'll + YnPAP-' = -1'22 - Y2yPA~'p-' 

 Incidentally, since Zo must be a symmetrical matrix the above equations 



