DECOMPOSITION METHODS FOR DETERMINING COMPLEX ONE-PARAMETER GENERALIZED INVERSE MOORE-PENROSE MATRICES (I)

Abstract

Analytical and numerical-analytical decomposition methods for determining complex one-parameter generalized inverse Moore-Penrose matrices are suggested. Analytical methods are based on the first two Moore-Penrose conditions. It is shown that in both cases the same solution to the problem is obtained, as it should be. Numerical-analytical methods are based on the obtained analytical relations and differential Pukhov transformations. In contrast to the known methods, in which we operate with rectangular generalized inverse matrices, in all the proposed analytical and numerical-analytical methods, we operate with square generalized (ordinary) inverse matrices. In view of this circumstance, an exact but not an approximate solution to the problem is obtained. A model example with a square matrix is considered, for which an analytical solution is obtained using two approaches - the determination of the usual inverse one-parameter matrix and the decomposition method for determining the same matrix. This matrix is defined in exactly the same way as numerical-analytical decomposition methods. In this paper, exact analytical and approximate numerical-analytical computational methods for determining complex one-parameter generalized inverse Moore-Penrose matrices are proposed. Numerical-analytical computational methods are easily implemented using modern information technologies, since they are based on simple recurrent computational procedures. It is assumed that all elements of the corresponding one-parameter matrices have a sufficient degree of smoothness at the center of approximation of solutions.