On Calculation of Pseudoinverse Square Matrix Based on Inversion

The paper focuses on the new formulae obtained for calculation of the Moore --- Penrose pseudoinverse matrix for a given quadratic matrix. The formulae are based on conversion as a sum of the given matrix and the exterior product of its left and right zero divisors by means of the inverse matrix. Such conversion allows us to use the standard calculation algorithms for pseudoinversion, improve equation conditionality in the case of badly scaled matrix pseudoinversion and complete pseudoinversion of symbolic matrixes. The paper exemplifies inversion of a low rank symbolic matrix. Finally, the set of corollaries from the theorems defining the pseudoinverse matrix calculation formulae is considered and a procedure of improving calculation accuracy of the pseudoinverse matrix is described

References

[1] Voevodin V.V., Kuznetsov Yu.A. Matritsy i vychisleniya [Matrices and calculations]. Moscow, Nauka Publ., 1984. 320 p.

[2] Bernstein D.S. Matrix mathematics. Princeton University Press, 2005. 768 p.

[3] Moore E.H. On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society, 1920, no. 26, pp. 394–395.

[4] Penrose R. A generalized inverse for matrices. Mathematical Proceedings of the Cambridge Philosophical Society, 1955, vol. 51, no. 3, pp. 406–413.

[5] Fredholm E.I. Sur une classe dequations fonctionnelles. Acta Mathematica, 1903, vol. 27, iss. 1, pp. 365–390. DOI: 10.1007/BF02421317

[6] Albert A. Regression and the Moore — Penrose pseudoinverse. Academic Press, 1972. 180 p.

[7] Zubov N.E., Mikrin E.A., Ryabchenko V.N. Matrichnye metody v teorii i praktike sistem avtomaticheskogo upravleniya letatelnykh apparatov [Matrix methods in theory and practice of automatic control systems of aircraft]. Moscow, Bauman MSTU Publ., 2016. 667 p.

[8] Yonglin Ch. The generalized Bott — Daffin inverse and its application. Linear Algebra and its Application, 1990, vol. 134, pp. 71–91. DOI: 10.1016/0024-3795(90)90007-Y

[9] Golub G.H., Van Loan Ch.F. Matrix computations. Johns Hopkins University Press, 2012. 784 p.

[10] Ikramov Kh.D. Chislennoe reshenie matrichnykh uravneniy [Numerical solution of the matrix equations]. Moscow, Nauka Publ., 1984. 192 p.