TRANSMISSION LINE EQUATIONS 



177, 



By differentiating the equation 



/(X)F(X) = A(X)/, 

 where A(X) is the determinant 



A(X) = |/(X) I = I GV + II\ + G'\, 

 it may be proved that 



GKAK~' + i7 + L'^ALG = L^'EK'' 



G'KA~'ir' + // + l-'a-'lg' = -L^'EK"' 



in which E is the diagonal matrix 



E = 



(A5.7) 



and 



(1) 

 Since the roots X, are assumed to be distinct, A(Xr) 9^ 0. 



We also have the equations 



KE^'L = L'E^'K' 



GKAE~'L - GL'A^'E~'K' = I 



The first equation of (A5.8) shows that KE~^L is a symmetrical matrix. 

 From this and the second equation it follows that 



(A5.8) 



GKAK~' - GL'A~'L' 

 From the first of equations (A5.7) and the second of (A5.6) 



GKAK~' - l~'a~'lg' = l~'ek~' 



(A5.9) 



(A5.10) 



and the comparison with (A5.9) shows that the matrix GL'A~ V is sym- 

 metrical. The other matrices of this type are also symmetrical as may now 

 be seen from equations (A5.6) and (A5.7). Results of this sort may be 

 obtained from physical principles by noting that Zo and Y o must be sym- 

 metrical matrices. 



