582 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



Bocher^ , for 



E\v) = A\v)A-\v)E{v)B-\v)B\v). 



Therefore it is also equivalent in the sense of par. 91 of Bocher \ (for 

 this is Theorem 1 of paragraph 94). Hence the normalized greatest 

 common factor of all fc-rowed minors of E^{p) is the same as that of 

 £'(p), that is, Eiip) • ■ • -E'tCp). But the greatest common factor of all k 

 rowed minors of E^i-p) is E?(p) • • • El{p), because of property (c). In 

 particular then £i(p) = £'S(p), and consequently Ek{v) = ^Jt(p) by 

 induction for I < k < R. Q.E.D. 



5.12 Definition: The normal form TF(p) of Z(p) is the matrix '^'^\p)E{p). 

 We write the elements of TF(p) in their lowest terms, 



W{p) = A{p)Z(p)B(p) = 



~ei(p) 

 ^i(p) 







• • • 

 -^iip) 





0. 



■•0 



(2) 



with the polynomials ek{p), ^k(p) each having unit leading coefficients. 

 5.13 Theorem: The normal form W(p) of Z{p), as given by (2), has the 

 properties (a'), (b'), (c'), (d'), and (e') listed below. Further- 

 more, any TF°(p), given by (2) wdth superscripts on W, A, B, Ck , and 

 ^;fe(l < k < R), which satisfies (a'), (b'), (c'); and (d') with correspond- 

 ing superscripts, is in fact W{p). 

 (a') R is the rank of Z(p) 

 (b') For each k, 1 < k < R, ek{p) and ^k{p) are relatively prime 



polynomials with unit leading coefficients, 

 (c') Each ek{p) is a factor of ek+i{p), I < k < R — 1, and each ^i(p) 



is a factor of ^j-i(p), 2 < j < R. 

 (d') A{p) and B{p) are polynomial matrices each with a constant 



non-vanishing determinant 

 (eO ^i(p) = ^z(p). 



Proof: (a') and (d') follow immediately from (a) and (d) of 5.1. (b') is 

 a matter of definition, (c') follows from (c) of 5.1 and the definition, 

 5.12, since the effect of cancelling common factors in each fraction of 

 the sequence 



E,{p) E,{p) EAp) 



■^z(p)''^z{py '" '^z(p) 



