544 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



2.16 If A and B are constant non-singular matrices, then 



8(Z) = 8(AZB). 



It is evident then that 8(Z) is a geometrical property, being constant 

 over the usual equivalence classes 



W'Z(p)W 

 or 



W-'Z{p)W 



of matrices. Hence we may speak of the degree 5(Z) of an operator Z(p). 



2.17 If Z(p) is formed from an m X m matrix Zi(p) by bordering the 

 latter with zeros, then 



8{Z) = 8{Z,). 



2.18 Concerning the degree 8{Z) we here state a fundamental theorem: 

 Theorem: The 2n-pole whose existence is asserted by 1.1 can be con- 

 structed with 8{Z) reactive elements, and no fewer. 



The proof of this theorem will be distributed through Sections 4 and 

 6. In fact, we must even define exactly the phrase "can be constructed 

 with X reactive elements." This will be done in Section 3. 



2.2 Lemma: If Zi{p) and Z-iip) are PR operators or matrices, then 



Z{p) = Zy{p) -f Z,{p) 



is also PR. If either of Zi{p) or Ziip) is non-singular, then Z{p) is. 



Proof: Clearly Z{p) is symmetric. By 2.09, therefore, Z{p) is PR if 

 the function 



{Z{p)h, k) = (ZMk, k) + (Z,(p)k, k) (1) 



is PR for each A; € K. The right hand side is obviously PR by hypothesis. 



If either of Zi(p) is non-singular, the function (1) cannot vanish in 



r^. unless A; = (this is 2.02). Hence in this case Z{p) also is non-singular. 



2.21 Clearly 2.2 is independent of the implication, tacit in the notation, 

 that the operators involved are impedances. The lemma holds for PR 

 operators, whether interpreted as operating from K to V (impedances) 

 or from V to K (admittances). % 



2.3 In (I, 6.21) and (I, 6.3) it was noted that any n X n impedance 

 matrix Z(p) defines by fiat a general 2w-pole N whose impedance matrix 

 is that Z(p). Such is the generality of the notion of general 2n-pole 

 (I, 4). 



