Dominant matrices and their application to network synthesis under topological constraints

Abstract

In this paper some algebraic properties of the Gaussian decomposition of a dominant 1 1 A matrix is said to be dominant if it is a symmetric matrix with real coefficients and if each of its main-diagonal elements is not less than the sum of the absolute values of all the other elements in the same row. matrix into a product of a lower and an upper triangular matrix, are discussed. It is shown that many properties of the parent dominant matrix are preserved in the derived triangular matrices. Moreover, around any such decomposition, A = T′T, with A—dominant and T—upper triangular, a family of dominant matrices, T′DT, may be defined. It is shown in the paper, that this method of generating new dominant matrices, proves to be of importance in the theory of electrical networks. The paper contains some examples of the application of this concept to synthesis of two-element-kind network functions if the realization is constrained to be achieved around a given one-element-kind subnetwork.