266 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



and, because Vm and V2 are real, vo = Vo , h = Vi , and these vectors are 

 real. Using similar reasoning for w, 



(vj) = ivi , YiP^i), (>', k) = {u, , Y(p)v,), (3) 



by 12.61. The equality {u, k) = (v, j) now follows from (3) and the duals 

 of 16.11, 14.11. Hence we have P5(A) for L and 12.0 is proved. 



16.7 We now prove an important 



Lemma: Let Z{p) be a PR operator from K to V. Let Tl be the set 

 of p where Z(p) is defined and has a rank equal to its nominal rank. Let 

 L be the correspondence with domain r^ and pairs 



[Z{p)k, k], keK, . 



Then L satisfies PI, • • • , P7. 



Proof: L satisfies PI and P2 (6.3). Tl satisfies P4 by the argument of 

 12.52. Then L satisfies P4, for by 16.11 



Z{p)k = Zip)k. 



L satisfies P5(I) by 14.11 and 16.11. r^ satisfies P6 by 12.52. Then L 

 satisfies P6(I) and P7(I) by 16.12. The fundamental lemma, 12.0, 

 now proves that L satisfies all the postulates. 



16.71 We call a correspondence satisfying all the postulates PR. 



16.72 Proof of 16.02: Suppose Re{Z{po)k, k) = for some po€T+ . 

 Because this function of p is PR (16.12) we have 



Jk(p) = {Z{p)k, k) ^ 0. 



Hence A;eKi = Kz.o (12.12, 12.55). Hence [0, k]eL(p) for every peVL. 

 That is 



Z{p)k = for peTL . 



16.73 Corollary: If Z{po)k = for some poeT+ , then Zip)k = 0. 

 For the hypothesis here implies that of 16.72. This is the analog of 

 15.12; the result of 16.02 is stronger. 



16.8 An important consequence of 16.7 is the 



Lemma*: If Z(p) is PR and of rank m, then there exists a real coordi- 

 nate frame in which the matrix [Z(p)] is an m X m nonsingular PR 

 matrix [Zi(p)] bordered by zeros. 



Proved by Cauer^. 



