150 



BELL SYSTEM TECHNICAL JOURNAL 



1.11 Proof of the Second Method 



To establish the second method we must prove the various formulas 

 which are used. These formulas all involve the square matrix N{yr) de- 

 fined by (1.23). 



Since A^(7r) is proportional to /^(7r) it follows that iV(7r) may be ex- 

 pressed as 



N{yl) = PrPr--- (1.59) 



where pr is the column matrix defined in Appendix I and pr is a row matrix 

 specified by pr and N{^r)- Applying Sylvester's theorem to the unit 

 matrix gives 



Pi 



/ = Z N{yl) = J^prPr-= [pl , p2 , ps] P2 



-.P3j 



where the two matrices on the extreme right are partitioned square matrices. 

 From the definition of P in Appendix / it follows that 



[pl , P2 , pi] = P, 



pl 



P2 

 LP3j 



= P-' 



(1.60) 



These relations enable us to verify the equations (1.24) when T is equal 

 to PGP~ . Thus for the first of equations (1.24) 



r = PGP ' = [p,,p,, p,]G 



Pl 



P2 

 LP3j 



= [pl , pi , pi] 



71 pl 



72 P2 

 _73P3_ 



= S PrJrPr = ^ N(yl)yr 



The second equation of (1.24) follows likewise from the expression (1.55) 

 for e~'' . 



The third equation of (1.24) follows at once from the first when we use 

 (1.45), Zo — rY~ . The fourth equation is obtained by writing 



e~'''^Zo = PM(x)P~'PGP~'V~' 



= PM{x)GP'''Y~^ 



and proceeding as in the case of the first equation. 



All of the other equations connected with the second method may be 

 proved in a similar way. Incidentally, the formulas obtained by the second 

 method are closely related to the "special form of solution" described in 

 §6.5 of F.D.C. 



