Abstract
There are various ways to introduce the generalized inverses. We introduce them by considering the problem of solving systems of linear equations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G.W. Stewart, Introduction to Matrix Computation (Academic Press, New York, 1973)
C.C. MacDuffe, The Theory of Matrices (Chelsea, New York, 1956)
N.S. Urquhart, Computation of generalized inverse matrtices which satisfy specified conditions. SIAM Rev. 10, 216–218 (1968)
S. Zlobec, An explicit form of the Moore-Penrose inverse of an arbitrary complex matrix. SIAM Rev. 12, 132–134 (1970)
A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, 2nd edn. (Springer Verlag, New York, 2003)
Y. Chen, Generalized Bott-Duffin inverse and its applications. Appl. Math. J. Chinese Univ. 4, 247–257 (1989). in Chinese
S.L. Campbell, C.D. Meyer Jr., Generalized Inverses of Linear Transformations (Pitman, London, 1979)
X. He, W. Sun. Introduction to Generalized Inverses of Matrices. (Jiangsu Science and Technology Press, 1990). in Chinese
Y. Chen, The generalized Bott-Duffin inverse and its applications. Linear Algebra Appl. 134, 71–91 (1990)
C.L. Lawson, R.J. Hanson, Solving Least Squares Problems (Prentice-Hall Inc, Englewood Cliffs, N.J., 1974)
Å. Björck, Numerical Methods for Least Squares Problems (SIAM, Philadelphia, 1996)
M. Wei, Supremum and Stability of Weighted Pseudoinverses and Weighted Least Squares Problems Analysis and Computations (Nova Science Publisher Inc, Huntington, NY, 2001)
L. Eldén, A weighted pseudoinverse, generalized singular values and constrained least squares problems. BIT 22, 487–502 (1982)
C. Bu, W. Gu, J. Zhou, Y. Wei, On matrices whose Moore-Penrose inverses are ray unique. Linear Multilinear Algebra 64(6), 1236–1243 (2016)
Y. Chen, On the weighted projector and weighted generalized inverse matrices. Acta Math. Appl. Sinica 6, 282–291 (1983). in Chinese
L. Sun, B. Zheng, C. Bu, Y. Wei, Moore-Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64(4), 686–698 (2016)
J. Ji, Y. Wei, Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product. Front. Math. China 12(6), 1319–1337 (2017)
C. Deng, Y. Wei, Further results on the Moore-Penrose invertibility of projectors and its applications. Linear Multilinear Algebra 60(1), 109–129 (2012)
D.S. Djordjević, P.S. Stanimirović, Y. Wei, The representation and approximations of outer generalized inverses. Acta Math. Hungar. 104, 1–26 (2004)
Y. Wei, H. Wu, On the perturbation and subproper splittings for the generalized inverse \(A_{T, S}^{(2)}\) of rectangular matrix \(A\). J. Comput. Appl. Math. 137, 317–329 (2001)
Y. Wei, H. Wu, (\(T\)-\(S\)) splitting methods for computing the generalized inverse \(A_{T, S}^{(2)}\) of rectangular systems. Int. J. Comput. Math. 77, 401–424 (2001)
Y. Wei, N. Zhang, Condition number related with generalized inverse \(A_{T, S}^{(2)}\) and constrained linear systems. J. Comput. Appl. Math. 157, 57–72 (2003)
M. Wei, B. Zhang, Structures and uniqueness conditions of MK-weighted pseudoinverses. BIT 34, 437–450 (1994)
J. Shao, H. Shan, The solution of a problem on matrices having signed generalized inverses. Linear Algebra Appl. 345, 43–70 (2002)
J. Zhou, C. Bu, Y. Wei, Group inverse for block matrices and some related sign analysis. Linear Multilinear Algebra 60, 669–681 (2012)
J. Zhou, C. Bu, Y. Wei, Some block matrices with signed Drazin inverses. Linear Algebra Appl. 437, 1779–1792 (2012)
G.W. Stewart, On scaled projections and pseudoinverses. Linear Algebra Appl. 112, 189–193 (1989)
M. Wei, Upper bound and stability of scaled pseudoinverses. Numer. Math. 72(2), 285–293 (1995)
O.M. Baksalary, G. Trenkler, Core inverse of matrices. Linear Multilinear Algebra 58(5–6), 681–697 (2010)
D. Rakić, N. Dinčić, D. Djordjević, Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)
Y. Wei, P. Xie, L. Zhang, Tikhonov regularization and randomized GSVD. SIAM J. Matrix Anal. Appl. 37(2), 649–675 (2016)
F.T. Luk, S. Qiao, Analysis of a recursive least squares signal processing algorithm. SIAM J. Sci. Stat. Comput. 10, 407–418 (1989)
F. Hsuan, P. Langenberg, A. Getson, The \(\{2\}\)-inverse with applications in statistics. Linear Algebra Appl. 70, 241–248 (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd. and Science Press
About this chapter
Cite this chapter
Wang, G., Wei, Y., Qiao, S. (2018). Equation Solving Generalized Inverses. In: Generalized Inverses: Theory and Computations. Developments in Mathematics, vol 53. Springer, Singapore. https://doi.org/10.1007/978-981-13-0146-9_1
Download citation
DOI: https://doi.org/10.1007/978-981-13-0146-9_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-0145-2
Online ISBN: 978-981-13-0146-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)