FORMAL REALIZAIULITV THEORY — I • 233 



in fact is non-degenerate in a sense similar to that of 5.7, for the current 

 amplitudes />• can be specified arbitrarily, and the resulting voltage 

 amplitudes v are then fixed by /,■ and />, by ((>). 



Z(p) is called the impedance matrix of the 2n-pole N/, . It is also 

 sometimes called the open-circuit, impedance matrix, because each 

 Zrsip) is, by (7), the voltage amplitude across (7V , Tr) when the current 

 amplitudes at all terminals save (7\ , T.,) are zero — i.e., when all pairs 

 sa\-e the s-th are on open circuit. 



(■).:U Dually, the pairs 



[v, Y(p)v] 



defined by an admittance matrix Y'(p) as v ranges over V define a linear 

 time inxariant 2n-pole which is non-degenerate. 



VII. WORK AND ENERGY 



7.0* A linear correspondence satisfying Pi and P2 is something which 

 abstracts the properties of linearity and time invariance. Most of the 

 remaining properties of physical networks involve the mention of work 

 or energy. These concepts enter our picture by way of the scalar product 

 ((', /.•) between a voltage /i-tuple (1) and a current /i-tuple (2), of 0.11. 

 This scalar product is defined by 



{V,k) = Zvrkr. (1) 



r=l 



7.01 If p = ice, one easily calculates from (3) and (4) of 6.11 that 



2ne{v,k) -y^rn^[ 



Z Vrm-rit) 



dt. 



That is, when p = iu, the real part of 2(v, A) measures the average total 

 power dissipated by the system of currents Av(/) against the driving 

 voltages Vr{t). 



When p is not a pure imaginary, the interpretation of the scalar 

 product {v, k) is not so clearly physical as this. The reader will ulti- 

 mately observe that our significant statements about such products can 

 all be reduced to statements applicable when p = ico, i.e., when the 

 power interpretation is valid. 



7.1* An important concept in what follows is that of the annihilator of 

 a linear manifold (Halmos^, par. 16). Let Vi C V be a linear manifold. 



* Technical paragraph as explained in Section 2.91. 



