Full-Rank Factorization and Moore-Penrose’s Inverse
C. MacDuffee apparently was the first to point out, in private communications, that a full-rank factorization of a matrix A leads to an explicit formula for its Moore-Penrose’s inverse A+. Here we apply this idea of MacDuffee and the Singular Value Decomposition to construct A+.
A. Ben-Israel, T. N. E. Greville, Generalized inverses: Theory and applications, Springer, New York (2003)
E. H. Moore, On the reciprocal of the general algebraic matrix, Bull. Am. Math. Soc. 26, No. 9 (1920) 394-395
R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc. 51 (1955) 406-413
A. Ben-Israel, The Moore of the Moore-Penrose inverse, Electron. J. of Linear Algebra 9 (2002) 150-157
R. E. Cline, Inverses of rank invariant powers of a matrix, SIAM J. Numer. Anal. 5 (1968) 182-197
G. V. Milovanovic, P. S. Stanimirovic, On Moore-Penrose inverse of block matrices and full-rank factorizations, Publications de L’Institut Mathématique, Nouvelle série, 62, No. 76 (1997) 26-40
C. Lanczos, Linear systems in self-adjoint form, Am. Math. Monthly 65, No. 9 (1958) 665-679
C. Lanczos, Extended boundary value problems, Proc. Int. Congr. Math. Edinburgh-1958, Cambridge University Press (1960) 154-181
H. Schwerdtfeger, Direct proof of Lanczos decomposition theorem, Am. Math. Monthly 67, No. 9 (1960) 855-860
C. Lanczos, Boundary value problems and orthogonal expansions, SIAM J. Appl. Math. 14, No. 4 (1966) 831-863
D. Kalman, A singularly valuable decomposition: The SVD of a matrix, The College Mathematics Journal 27 (1996) 2-23
C. Lanczos, Linear differential operators, Dover, New York (1997)
V. Gaftoi, J. López-Bonilla, G. Ovando, Singular value decomposition and Lanczos potential, in “Current topics in quantum field theory research”, Ed. O. Kovras, Nova Science Pub., New York (2007) Chap. 10, 313-316
I. Guerrero-Moreno, J. López-Bonilla, L. Rosales-Roldán, SVD applied to Dirac supermatrix, The SciTech, J. Sci. & Tech. (India), Special Issue (2012) 111-114
G. Bahadur-Thapa, P. Lam-Estrada, J. López-Bonilla, On the Moore-Penrose generalized inverse matrix, World Scientific News 95 (2018) 100-110
T. N. E. Greville, Some applications of the pseudoinverse of a matrix, SIAM Rev. 2, No. 1 (1960) 15-22
H. Yanai, K. Takeuchi, Y. Takane, Projection matrices, generalized inverse matrices, and singular value decomposition, Springer, New York (2011) Chap. 3
Copyright (c) 2018 MathLAB Journal
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors retain the copyright of their manuscripts, and all Open Access articles are distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided that the original work is properly cited.