The study of the properties and the construction methods of pseudoinverse matrices forms the fundamental basis of control theory and systems analysis, having significant applications across numerous applied problems. The definition of the pseudoinverse matrix was first proposed at the beginning of the 20th century by the mathematician E.H. Moore. Later, independently of Moore and in a slightly different form, the pseudoinverse matrix was defined and investigated by the English mathematician Roger Penrose and other authors. The statement regarding the existence and uniqueness of a pseudoinverse for any matrix over real or complex numbers is known as the Moore - Penrose theorem. T. Greville proposed recurrence formulas for calculating pseudoinverse matrices in the case of expanding a system of equations. Consequently, the class of pseudo-inversion problems can be extended to adaptive data processing tasks.

The theory of pseudo-inversion was further developed in the works of domestic and foreign scientists. Notably, the research of M.F. Kyrychenko and his students and followers - Yu.V. Krak, F.H. Garashchenko, and V.S. Donchenko - should be highlighted. They established a powerful scientific school in the fields of systems analysis, control theory, pattern recognition, gesture synthesis, and intelligent control. The main research directions of this school include perturbation theory of pseudoinverse matrices, constructive methods, and specific algorithms (recurrence formulas) that can be implemented for real-world technical systems.

This paper proposes new mathematical results in the classical theory of matrix pseudoinversion for cases where the initial matrix is expanded or narrowed by a matrix of corresponding dimensions. The obtained results include generalizations of Greville's formulas for the case of matrix addition (expansion) and M.F. Kyrychenko's formulas for the case of matrix removal (narrowing). The results are presented in the form of theorems with corresponding proofs.

[1] Ben-Israel A., Greville T. N.E. Generalized Inverses: Theory and Applications. 2nd ed. Springer, New York, 2003.

[2] Albert A. Regression, pseudoinversion, recurrent estimation / Trans. with English M.: Nauka, 1977. 305 p.

[3] Cline R.E. Representations for the generalized inverse of partitioned matrix. SIAM J. Appl. Math. 32, 1964. 588-600.

[4] Kirichenko N.F., Lepecha M.P. Pseudoinverse and projective matrices perturbation in linear and nonlinear identification problems. Problemy Upravleniya I Informatiki (Avtomatika). Iss. 1. 2001. 6-22.