22 CHARACTERISTIC-NOISE MATRIX [Ch. 3 



The stationary conditions and corresponding values for p may be ob- 

 tained from the solution of the equivalent problem of determining the 

 stationary values of x^Bx, subject to the constraint x^Ax = constant. 

 Therefore, introduction of the Lagrange multiplier 1/X and recognition 

 that p may be regarded as a function of either the set of Xi or the set of 

 Xi* lead to the conditions 



d 

 dXi" 



U^Bx - - xUx j = 0, i = \,2,--',n (3.11) 



or simply 



Ax - XBx = (A - XB)x = (3.12) 



The values of X are then fixed by the requirement 



det (A - XB) = = det (B-^A - XI) (3.13) 



where 1 is the unit matrix. This means that the values of X are just the 

 eigenvalues of the matrix B~^A = —\{^ + Z''^)~^EE''^. 



The matrix x yielding any stationary point of p must satisfy Eqs. 3.12, 

 as well as the constraint x^Ax = constant. Let Xs be one eigenvalue of 

 B~^A and x^^^ be the corresponding solution (eigenvector) of Eq. 3.12. 

 Then premultiplication of Eq. 3.12 by x^®^"*" yields 



x(s)tAx(s) = x^x('^"^Bx(') 

 or 



which is real and equal to the corresponding stationary value of p. 



It follows that the stationary values of the exchangeable power Pe,i are 

 the negatives of the (real) eigenvalues of the matrix 



AB-i = -i(Z + Z+)-iEE+. 



We therefore define: 



Characteristic-noise matrix = N = - J(Z + Z+)~^EET (3.15) 



and conclude that: 



The stationary values of the exchangeable power Pe,i are the negatives of 

 the {real) eigenvalues of the characteristic-noise matrix N. 



