FORMAL REALIZARILITY THEORY — -I 203 



KkOC) If Zip) has a polo at ?j = =o , that i^olo is simple and 



Zip) = ph' + Z,ip) 



whoro R = ir = R, h' > and Z^ip) is I'K. 



J 0.07 There is property of rational scalar functions f{p), whether PR 

 or not, that is essential in the Brune theory: the existence of a finite 

 integer, the degree of /. Each step in the Brune reduction of f(p) leaves 

 an unreduced portion which is of lower degree than the function upon 

 which the step was performed. The finiteness of the original degree of / 

 then guarantees the termination of the process in finitely many steps. 

 There exists jjilso for rational matrices (and operators) a concept of 

 degree. This degree plays the same role in the general Brune process for 

 2 /i -poles as the degree of a scalar function does in the process for 2-poles. 

 To define this degree and develop its properties requires an excursion 

 into classical algebra. Since we shall not need these ideas until Part II 

 we defer further discussion of them to that part. 



10.1 If Z(p) is PR it follows at once that the matrix [Z{p)] is PR in any 

 real frame. 



Proof: Two such matrices are related by 



[Zip)], = [U][Z(p)][UY 



where U is real, by 11.22 and the argument in 8.0. The PR properties 

 of [Z(p)] are obviously preserved by this operation. 



10.11 If Z(p) is PR, then 



Zip) = Z'ip) = Z*(p) = Zip). 

 Proof: Use 10.0 and 14.03 in a real frame. 

 10.12 If Zip) is PR, then for any given A'eK the function 



Juip) = iZ(p)k, k) 



is a PR scalar function. It follows that the limitation in (iv) of 10.0 

 to real k is a simplification, not a restriction. 



Proof: J kip) is independent of coordinate representation. By use of a 

 real frame, (i) of 10.0 implies (i) of 15.0. 



Bv 14.01 and 10.11 



Jkiv) = {Z*ip)k, k) = iZip)k, k) = J,ip). 

 This is (ii) of 15.0. For any k, 14.12 and (iv) of 10.0 imply (iii) of 15.0. 



