FORMAL REALIZABILITY THEORY — II 579 



Using the fact that M{p) is determined by its quadratic form, we see 

 that M^ is the matrix whose form is the residue of that of M{p) at 

 p = CO , This residue was computed in (27) of 4.65 to be 



2{G-'h, h) + 4{Fh, h) (1) 



when the current vector, (17) of 4.58, is 



ii e A; © J2 , (2) 



and, (25) of 4.65, 



2h =j,-{-j\- 2GB(jr + k). (3) 



Here ji , ^2 e J and k eKi . 



Now M„ is an (w + m) X (« + m) matrLx by construction. Then 



V = n -\- m — rank (M„) (4) 



is its nuUity, the dimension of its null space. This is proved in Halmos^, 

 par. 37, for dimensionless operators, and a similar proof applies to im- 

 l)edance operators. 



Now for any symmetric and semidefinite impedance operator Z, the 

 null space of Z is exactly the aggregate of currents k such that the 

 quadratic form 



{Zk, k) = 0. 



This may be seen at once by choosing a coordinate frame in which the 

 matrix of Z is diagonal. Since we know from 4.65 that AI^ is symmetric 

 and semidefinite, we can compute v as the dimensionality of the space 

 of vectors (2) above for which (1) vanishes. 



As noted in 4.65, he], and (1) vanishes if and only if /i = 0, because 

 G~ , as an operator from J to U, is definite (semidefinite and non-singu- 

 lar). Hence v is the maximum number of linearly independent vectors 

 (2) for which, from (3), 



(1 - 2GB)j\ + j, - 2GBk = 0. (5) 



The left member of (5) is a vector in J depending linearly and homo- 

 geneously on the vector (2). Hence, regarding J as a subspace of the 

 space J © Ki © J in which (2) lies, the left member of (5) is the value 

 in J © Ki © J of a certain linear operation applied to the vector (2), 

 Let us cull this operator P. The numl)er v, by definition the number of 

 linearly independent vectors (2) for which (5) holds, is the nullity of P. 

 The dimension of P is ?i + m, and its rank is clearly ni because the left 

 member of (5) — a typical element in the range of P — lies in J and by 



