580 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



suitable choice of j-> can be made to be any element of J. Hence the 

 nullity of P is (n + m) — m = n (Halmos^, par. 37). That is 



V = n, 



and, by (4) 



rank (M^) = m. 



A parallel argument Avill establish the same result for J/o . Hence 



8(M) = 2m = 5(N) - 5(Ni) 



by a result of 4.7. Therefore Mad and Mad can be reahzed with 



6(N) - 5(N,) 



reactive elements and 4.07 holds for IB. 



V. THE DEGREE OF A RATIONAL MATRIX 



5.0 In this .section we consider arbitraiy n X n matrices Z(p) whose 

 elements are rational functions of the complex variable p. They are 

 treated, generally, as arrays of functions with certain rules for addition, 

 multiplication, and reciprocation, wdthout geometric interpretation. A 

 geometric development is possible, but would be cumbrous. Related 

 ideas may be found, geometrically developed, in Appendix I of Halmos". 



This section deals w^holly Avith concepts well known in the algebraic 

 theory of matrices over an arbitrary field — in this case the field of 

 rational functions. I have not found, however, any place where the 

 particular developments which seem to be needed here are made suffi- 

 ciently explicitly for reference. Accordingly, the presentation here is 

 somewhat detailed. The particular path of argument followed is only 

 one of many possible; it was chosen to lead easily to results needed in 

 Section 6, and to parallel generally the rest of the paper. 



This section could be abbreviated somewhat if one restricted himself 

 to PR matrices Z(p). We prefer not to limit the applicability of these 

 results, however, since they may well be useful in non-passive realiza- 

 bility theoiy. 



5.01 Definition: If R{p) is a rational function of the form 



Rip) = (p - p,y"Ri(p), 



where Ri{p) is finite and not zero at po , and w may be of an}^ sign, we 

 call m the exponent of (p — po) in R(p). The number 



