FORMAL REALIZABILTTY THEORY — II 557 



the theory of qu:uhatic forms. Fix co and eoiisider the matrix 



/i'(/a)) - \S 

 as a function of X. Its (letermiuaut, 



A,(X) = I R{ioo) - \S\, 



is an n degree polynomial in X with the following two properties: 

 (a) The coefficient of X" in Au(K) is not zero and is indepencknit of co, 

 (l3) The n roots of 



A„(X) = (2) 



are real and positive. 



Now R(i(j:) is rational, hence continuous, and finite for all oj, including 

 oj = °o, by the hypothesis that N is in (1, 1). By (a) above, therefore, 

 each root of (2) is a continuous function of oj on the compact set — <» 

 < CO < <x>. Let a(co) denote the least root of (2). Then a(w) is again 

 l)ounded and continuous for all co. There is, therefore, an coo where a(a)) 

 takes its least value. This is the wo referred to in the lemma, and 



a = a(ajo). 



We see that this calculation requires solving an n degree polynomial 

 equation containing a parameter (w), and then minimizing the least root 

 b}' varying the parameter. Though some properties of R{ico) are available 

 to assist in the process, and the choice of S is somewhat free to us, this 

 is scarcely a feasible calculation in practice. Even w^hen one reduces the 

 minimizing problem to finding the roots of a derivative, there remains a 

 prodigious calculation in all but the simplest cases. 



Since by its definition i?(tco) = R( — io:), we may take coo > 0. 



The relation (1) above implies that 



{R{ico)k, /v) > 



for all real co and all k e K, because Z(p) is PR. That is, R{i(xi) is semi- 

 definite. The hypothesis that Z(p) has no zero of its real part R(j)) on 

 p = ioi then implies that R{iui) is positive definite. All of (i), (ii), {a), 

 and (/3) then follow from well-known properties of definite quadratic 

 forms. They may, for example, all l)e deduced from Halmos , paragraphs 

 62, 63, and 74, by choosing a coordinate frame in which the operator 

 corresponding to *S above is represented by the unit matrix. A more 

 elegant reduction to the cited results of Halmos can also be constructed. 



4.32 Lemma: Given N in (1, 1), we choose any real constant symmetric 



