176 BELL SYSTEM TECHNICAL JOURNAL 



and we shall suppose these roots to be distinct. Let the column kr and the 

 row Ir be such that 



krlr = F{\r) (A5.2) 



where /^(X) is the matrix adjoint to/(X), and let the square matrices K and 

 L be defined by 



K = [ki,k2, • • • kr,], L = 



(A5.3) 



Comparison of (A5.3) and (2.50) suggests that when G and fi^ are expressed 

 in terms of the F's we have the relations 



K = P, L = P' (A5.4) 



The method of choosing the column Pr and the row pr shows that they are 

 related by 



prVr = yrF{\r) 



instead of (A5.2) where 7^ may turn out to be any non-zero constant, and 

 consequently equations (A5.4) are not satisfied in general. Nevertheless K 

 and L may be regarded as particular choices for P and P' . In the same way 

 A' and L may be regarded as particular choices for Q and Q' when G and H 

 are expressed in terms of the Z's. There is still some arbitrariness connected 

 with K and L since the product krh is unchanged when the kr is multiplied 

 by some number and Ir is divided by the same number. 

 The relations which correspond to (2.52) and (2.57) are 



GKK^ + HKA -\- G'K = 



(A5.5) 

 GL'A'^ + HL'A~' + G'L' = 



where A is the diagonal matrix whose elements are Xi , X2 , • • • X^ . These 

 relations are consequences of the properties of kr and Ir- Two more rela- 

 tions may be obtained by transposition. From the first of (A5.5) and the 

 transposed of the second it follows that 



GKAK~' -\- H + G'KA~^K~^ = 



(A5.6) 

 L'^ALG -\- H + L-'A-'LG' = 



where it is assumed that the reciprocal matrices K~ and L exist. Com- 

 binations similar to KAK~^, KA~^K~^, etc. enter the expressions (2.59) 

 for Zo and Yq. 



