FORMAL REALI7ABILITY THEORY — II oG9 



n-tuple of the form 



w e (18) 



where u e U. Regarding (15), Avith this determination of Vo , as simply 

 an m-tuple m (ignoring its last n - m zero components), we see that (17) 

 and the ordered pair (14), (15) are now currents and voltages in a 

 2(n + m)-pole Mad obtained from Mad by shorting and thereafter ignor- 

 ing the lower n - m terminals on D-D. 



■4.59 Now (17) is unrestricted. Given it, the corresponding voltages 

 can be computed from (14) and (15) by determining /'n so that (15) lies 

 in U. Hence Mad has an impedance matrix, since any single valued 

 linear mapping from (17) to voltages can be described by a matrix. Oiu- 

 job is now to show that this matrix comes under 3.1. Before doing this, 

 however, we shall point out that a realization of Mad provides one 

 for Mad . 



Fig. 7 shows how a 2(2n)-pole equivalent to Mad would be con- 

 structed from Mad • The equivalence is evident almost at once: The 

 pairs of Mad are the currents (17) and the voltages (14) and (15) with a 

 special determination of v^ , where (15) is regarded as an m-tuple. The 

 current (10) is clearly that which flows in the 2(2n,)-pole of Fig. 7 when 

 (17) flows in Mad • Furthermore, regarding (15) as an «-tuple of the 

 form (18), we see that the voltages in Fig. 7 can be obtained from (14), 

 (15) b}'^ adding an arbitrary voltage of the form 



(0 e v) e (0 e v), 



where v e Vi of course. This arbitrarj'' added voltage eliminates the 

 special role played by Vq in (14) and (15). Hence therein Vo itself may be 

 considered to be an arbitrary element of Vi , and (14), (15) represent the 

 \oltages in Fig. 7. The pairs admitted by the 2(27?)-pole of Fig. 7 are 

 therefore exactly those admitted by Mad , Q.E.D. 



4.60 We have now established that Mad bas an impedance matrLx, say 

 M{p). M{p) operates from an (n + w) space of currents (17) of 4.58 

 to an (n + ?n) space of voltages (14), (15) of 4.56, where in (15) we prop- 

 erly choose Vo so that the last (n — m) components are zero and can 

 he ignored. 



Now any impedance matrix Z(p) is completely determined when we 

 know for each two currents m\ and m^ the scalar product 



{Z(p)m, , m.-) (1) 



(Cf. Halmos'', par. 53). We shall make this computation for M(p). The 



