Sec. 2.1] 



IMPEDANCE-MATRIX REPRESENTATION 



11 



with identical statistical properties. Here, as in the rest of this work, we 

 retain only positive frequencies. In order to preserve the same multipli- 

 cative factors in power expressions for both random and nonrandom 

 variables, we shall depart from convention by using root-mean-square values 

 for all nonrandom complex amplitudes. 



A convenient summary of the power spectral densities is the matrix 



EEt = 



_EnEi 



EnEi 



EnEk* 



F F * 



(2.1) 



where the superscript dagger indicates the two-step operation composed 



of forming the complex conjugate of and transposing the matrix to which 



it refers. Briefly, A+ is called the Hermitian conjugate of any matrix A.^ 



The matrix EE^ is its own Hermitian conjugate, because EiEk* = 



Ek*Ei = (EkEi*)*. Such a matrix is said to be Hermitian. 



In addition, we can show that EE^ is a positive definite or semidefinite 

 matrix. Construct the real nonnegative quadratic form : 



(Xi*Ei-\- . . . +Xi''Ei+ . . . -{-Xn^'En) (a:i£i*-i-. . . + XiEi^'-h . . . +Xr,En*) >0 



where the Xi are arbitrary complex numbers. In terms of the column 

 matrix 



'~Xi 



X = 



and the column matrix E, the foregoing expression becomes 



xtE(x+E)+ = x+EE+x > 



^ F. B. Hildebrand, Methods of Applied Mathematics, Prentice-Hall, New York 

 (1952). 



