174 BELL SYSTEM TECHNICAL JOURNAL 



These are partitioned matrices. The square matrices have 2w rows and 

 columns and the column matrices have 2ni elements. The first of these 

 relations may be written as 



[I; !:][;::: a = K a 



where / denotes the unit diagonal matrix of order m. Multiplying the two 

 matrices on the left together and equating the elements of the product 

 to the elements on the right gives 



ZiiFu + Zi2Fi:i = / 



ZnYn + Z12F22 = 



(A4.3) 

 Z2xYn + Z22F21 = 



•2^2lFi2 + Z22F22 = / 



Transposing the matrices in these equations leads to other relations. Thus, 

 from the first we obtain YnZn + F12Z21 = /. These equations also yield 

 expressions for the F's in terms of the Z's and vice versa. 



A somewhat similar treatment involving equations (2.1) and (2.3) leads 

 to expressions for the Z's in terms oi A, B,C and D. The F's may be like- 

 wise expressed. These relations are given in the following table. 



Fii = DB I'll = Zu — Z12Z22 Z21 



F12 = C — DB A = — B F12 = Z21 — Z22Z12 Zii 



F21 = —B F21 = Z12 — Z11Z21Z22 



F22 ^^ B A F22 = Z22 — Z2iZn Z12 



Zu = AC-' ZTi = Fu - Fi2F2tF2i 



Z12 = ac~'d - b = c'"' zr2' = F21 - F22Fr2'Fu 



Z21 = C Z21 = F12 — F11F21 F22 



^22 = C-'D Z72 = F22 - F2iFri'Fi2 



A = Z11Z21 = — F21 F22 



(A4.4) 



B = ZwZix Z21 — Z12 = — F21 



C = Zt = Yu - FiiF7i'F22 



D = Z21 Z22 = — F11F21 



AD' - BC = I AB' = BA' 



CD' = DC DA' - CB' = I 



\A B'Y' _\ D' -B'^ 

 [C d\ \_-C' A' J 



r v°{n) ^ _\A Si r u\n) 1 

 l-nn)\ \C D\l-f{n)\ 



