DETERMINING COMPLEX ONE-PARAMETER GENERALIZED INVERSE MOORE-PENROSE MATRICES (II)

Abstract

Analytical and numerical-analytical decomposition methods for determining complex one-parameter generalized inverse Moore-Penrose matrices are presented. Analytical methods are based on the 4th Moore-Penrose condition. Three options of the analytical solution are presented. The 1st option is based on complex decompositions of the given matrix and its Moore-Penrose inverse matrix. The 2nd option is based on a combination of the 1st and the 4th Moore-Penrose conditions. The 3rd option is based on a combination of the 2nd and the 4th Moore-Penrose conditions. In the case of analytical solution option 1, if any of the obtained iteration procedures converge, the Moore-Penrose inverse matrix can be determined using the corresponding matrix blocks. In the case of analytical solution options 2 and 3, the obtained relations are obviously simple compared to the iterative procedures of option 1, which leads to the direct determination of the Moore-Penrose inverse matrix. Numerical-analytical methods are based on the obtained analytical solution options 2 and 3, and use differential Pukhov transformations as a primary mathematical tool.

A model example with a square matrix is considered, for which a precise numerical-analytical solution is obtained using matrix discretes. Based on these discretes, the corresponding matrix blocks were restored and the Moore-Penrose inverse matrix was obtained according to its complex decomposition.