Definition of Complex One-Parameter Generalized Moore–Penrose Inverses Using Differential Transformations

This study presents analytical and numerical-analytical decomposition methods for determining complex one-parameter generalized inverse Moore–Penrose matrices. The analytical approach is based on the third Moore–Penrose condition, offering three solution options. The first option employs complex decompositions of the given matrix and its Moore–Penrose inverse. The second option combines the first and third Moore–Penrose conditions, while the third option integrates the second and third conditions. For the first and third options, if any derived iterative procedure converges, the Moore–Penrose inverse matrix can be constructed using the corresponding matrix blocks. In contrast, the second option provides simplified relations, enabling the direct computation of the Moore–Penrose inverse matrix. Numerical-analytical methods build on the second analytical solution, utilizing differential Pukhov transformations as the primary mathematical tool. A model example featuring a rectangular complex matrix is analyzed. A numerical-analytical solution is derived using three matrix discretes, from which corresponding matrix blocks are reconstructed. The Moore–Penrose inverse matrix is then obtained through its complex decomposition.