584 THE BELL .SYSTEM TECHNICAL JOURNAL, MAY 1952 



5.16 Corollarij: If C(p) and D{p) are polynomial matrices with constant 

 non-vanishing determinants, then the normal forms of Z{p) and 

 C{p)Z(p)D(p) are the same. 

 Proof: 



AZB = (AC~')CZD(D~'B) 



and the bracketed factors are again polynomial matrices with constant 

 non-vanishing determinants. 



5.2 Definition: The point po is a pole of Z(p) if some element of Zijp) 

 has a pole at p = po • If Po is not a pole of Z(p), we saj^ that Zfpo) is 

 finite, or that Z(p) is finite at po . 



5.21 If po is a pole of Z(p), we may expand each element of Z in partial 

 fractions and collect those terms having poles at po , obtaining, when 

 Po ?^ °°, 



z{v) = (v - p^rzr + (p - p,r^'Zr-i 



+ • • • + (p - Po)"% + Zo(p), 



where Zo(po) is finite, Zr 9^ 0, and the Za , 1 < A' < r, are matrices of 

 constants. If po = <» , we read p' for (p — po)" in (1), 1 < t < r. All 

 of Zo{p), Zi , • ■ • , Zr are uniquely defined by their construction from 

 Zip). 



5.22 Definition: If Z(p) is given by (1) above, then r is the order of the 

 pole of Z(p) at Po . 



5.23 Clearly, if Z(p) has the form (1) at po ^ ^ , some non-vanishing 

 element of Z{p) has a denominator containing the factor (p — poY, and 

 no element has a pole of order higher than r at po . Hence (p — poY 

 di\ddes ^z(p), but no higher power of (p — po) does. Therefore, by (e') 

 of 5.13, the normal form ir(p) of Z{p) has a first element with an r* 

 order pole at po . In particular, then, po 5^ <» is a pole of order r of 

 Z(p) if and only if it is a pole of order r of Tr(p). 



5.24 Definition: Consider a pole of order r of Z(p), say po , with po ?^ <» . 

 In the normal form W(p) of Z(p), (2) of 5.12, let 7a- be the order of the 

 pole of the A;' diagonal element 



efc(p) 

 ^a(p) 



at the point p = po . Then 7a > jk+i , and 71 = r. We write the 7a- in 

 an ordered array 



S{Z, Po) = [71 , 72 , • • • , 7«]- 



