Abstract
We prove that \({\rm{cond}}_{\dagger}(T)-1\leq {\rm{cond}}^{[2]}_{\dagger}(T)\leq{\rm{cond}}_{\dagger}(T)+1\), where T is a linear operator in a Hilbert space, \({\rm{cond}}_{\dagger}(T)\) is the condition number of computing its Moore–Penrose inverse, and \({\rm{cond}}^{[2]}_{\dagger}(T)\) is the level-2 condition number of this problem.
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Acknowledgements
The authors would like to thank Prof. Manjunatha Prasad Karantha and the referee’s detailed comments, which improved the presentation of this paper. Also, we would like to thank Prof. F. Cucker for useful discussions. H. Diao is supported by the Youth Foundation of Tianyuan Mathematics, National Natural Science Foundation of China, under grant 10926107, the National Natural Science Foundation of China under grant 11001045, Specialized Research Fund for the Doctoral Program of Higher Education of MOE, China, under grant 20090043120008, and by Training Fund of NENU Scientific Innovation Project of Northeast Normal University under grant NENU-STC08009 and by Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT). Partial work was finished when H. Diao visited Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University) in 2012. Y. Wei is supported by the National Natural Science Foundation of China under grant 11271084, Doctoral Program of the Ministry of Education under grant 20090071110003, 973 Program Project under grant 2010CB327900, Shanghai Education Committee, and Shanghai Science & Technology Committee.
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Diao, H., Wei, Y. (2013). On the Level-2 Condition Number for Moore–Penrose Inverse in Hilbert Space. In: Bapat, R., Kirkland, S., Prasad, K., Puntanen, S. (eds) Combinatorial Matrix Theory and Generalized Inverses of Matrices. Springer, India. https://doi.org/10.1007/978-81-322-1053-5_13
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DOI: https://doi.org/10.1007/978-81-322-1053-5_13
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