Definiteness of real and imaginary parts of complex electric network matrixes

Abstract

To solve a number of problems of optimization of electric power problems, it is necessary to know the definiteness of electric network matrices that are incorporated into quadratic forms of target functions and define their properties. To ground matrix definiteness, two criteria are used. One of them is based on finding proper matrix values, and the other is based on finding the sequence of the main minors of the determinant of a real symmetric matrix. A theoretical underpinning of the definiteness of the real and imaginary parts of complex electric network matrices using the principal minors of the matrix determinants is suggested. For positive matrix definiteness, it is necessary and sufficient to meet Sylvester’s criterion for positive definiteness, and for negative definiteness it is sufficient to implement the inequality with a number of negative terms which are the determinants’ principal minors. Proofs are given for the real and imaginary parts of the complex matrices of node impedances. The proof is based on the method of finding the sequence of the determinants’ main minors, in which the value of the previous main minor is used in calculating the following main minor.