542 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



although in terminology, notation, and method a direct continuation 

 of it. References to sections or paragraphs in Part I will be made thus: 

 (I, 6) or (I, 6.23). 



1.5 The distinction emphasized in Part I between operators, as abstract 

 geometrical objects, and matrices as concrete arrays of numbers repre- 

 senting these geometrical objects, is not one which we have now to 

 maintain with any strictness. We shall generally preserve it verbally 

 but not use the bracket notation for matrices introduced in Part I. 



II. PROPERTIES OF POSITIVE REAL OPERATORS AND MATRICES 



2.0 We recall that an impedance operator Z{p) is a linear function from 

 the linear space K of current vectors k to the linear space V of voltage 

 vectors v. A positive real operator Z{p) is one whose matrix in any real 

 coordinate frame is positive real. In Section 16 of Part I the following 

 properties of a PR operator Z(p) were established: 



2.01 Zip) has no poles in r+ .* 



2.02 If Rc(Z(p)k, k) = for some peT+, then Z(p)k ^ for all p. 



2.03 If it exists, Z~'(p) = Y(p) is PR. 



2.04 If Z(p) has a pole at p = po , it has one at p = po . 



2.05 If Z(p) has a pole at p = fcoo , that pole is simple and 



Zip) = ^^2 R + Zrip), 



where R is real, symmetric, semidefinite, and not zero, and Ziip) is PR- 



2.06 If Zip) has a pole at p = oo , that pole is simple and 



Zip) = pR + Z,ip) 

 where R and Ziip) are as in 2.05. 



2.07 It was emphasized at several points in Part I that the fact of pos- 

 sessing an impedance matrix, and that of possessing an admittance 

 matrix, are each restrictions on a 2n-pole N. It is readily verified from 

 (I, 6.3) and (I, 6.31) — and, indeed, well known — that if N has both an 

 impedance matrix Zip) and an admittance matrbc Yip), then 



Yip) = Z-\p). 



r+ is the open right half plane: all finite p such that Re{p) > 0. 



