The Pseudoinverse of a Rectangular or Singular Matrix and Its Application to the Solution of Systems of Linear Equations

Previous article Next article The Pseudoinverse of a Rectangular or Singular Matrix and Its Application to the Solution of Systems of Linear EquationsT. N. E. GrevilleT. N. E. Grevillehttps://doi.org/10.1137/1001003PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout1 R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406–413 MR0069793 0065.24603 CrossrefGoogle Scholar2 R. Penrose, On best approximation solutions of linear matrix equations, Proc. Cambridge Philos. Soc., 52 (1956), 17–19 MR0074092 0070.12501 CrossrefGoogle Scholar3 Arne Bjerhammar, Application of calculus of matrices to method of least squares with special reference to geodetic calculations, Trans. Roy. Inst. Tech. Stockholm, 1951 (1951), 86 pp. (2 plates) MR0055793 Google Scholar4 Arne Bjerhammar, Rectangular reciprocal matrices, with special reference to geodetic calculations, Bull. Géodésique, 1951 (1951), 188–220 MR0043758 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails H∞ -PID distributed control for output leader-tracking and containment of heterogeneous MASs with external disturbancesJournal of Control and Decision, Vol. 9, No. 4 | 24 December 2021 Cross Ref Improving MRAC Transient Response Using Generalized Dynamic Inversion and Nullspace ParametrizationIFAC-PapersOnLine, Vol. 55, No. 12 | 1 Jan 2022 Cross Ref Successive Generalized Dynamic Inversion Control of Lateral Fighter Aircraft Dynamics*2021 9th International Conference on Systems and Control (ICSC) | 24 Nov 2021 Cross Ref Experimental noise source identification in a fuselage test environment based on nearfield acoustical holographyCEAS Aeronautical Journal, Vol. 12, No. 4 | 18 August 2021 Cross Ref A Model-free Deep Reinforcement Learning Approach for Robotic Manipulators Path Planning2021 21st International Conference on Control, Automation and Systems (ICCAS) | 12 Oct 2021 Cross Ref Some additive results on M-P inverse of operator productAsian-European Journal of Mathematics, Vol. 14, No. 01 | 18 October 2019 Cross Ref An optimal distributed PID-like control for the output containment and leader-following of heterogeneous high-order multi-agent systemsInformation Sciences, Vol. 541 | 1 Dec 2020 Cross Ref Mutual Direct State Controllability Analysis of Multivariable Underactuated LTI Systems2020 Australian and New Zealand Control Conference (ANZCC) | 26 Nov 2020 Cross Ref Using prior parameter knowledge in model‐based design of experiments for pharmaceutical productionAIChE Journal, Vol. 66, No. 11 | 26 August 2020 Cross Ref Mutual Direct Controllability of MIMO Systems and Application to Longitudinal Aircraft Control2020 24th International Conference on System Theory, Control and Computing (ICSTCC) | 8 Oct 2020 Cross Ref Neural Network-based Hybrid Position/Force Tracking Control for Flexible Joint Robot2020 IEEE International Systems Conference (SysCon) | 24 Aug 2020 Cross Ref Femtosecond-resolved observation of the fragmentation of buckminsterfullerene following X-ray multiphoton ionizationNature Physics, Vol. 15, No. 12 | 23 September 2019 Cross Ref Canonical Kane's Equations of Motion for Discrete Dynamical SystemsAIAA Journal, Vol. 57, No. 10 | 1 Oct 2019 Cross Ref Sequential model-based A- and V-optimal design of experiments for building fundamental models of pharmaceutical production processesComputers sequential biased estimation for nonorthogonai problemsJournal of Statistical Computation and Simulation, Vol. 3, No. 3 | 1 Jan 1975 Cross Ref Control of a binary distillation column utilizing a simple control lawThe Canadian Journal of Chemical Engineering, Vol. 52, No. 6 | 1 Dec 1974 Cross Ref MODEL REFERENCE ADAPTIVE CONTROL BASED ON LIAPUNOV'S DIRECT METHOD Part I: Theory and Control System Design†Chemical Engineering Communications, Vol. 1, No. 3 | 21 May 2007 Cross Ref Geometric Programming: A Technical State-of-the-Art SurveyA I I E Transactions, Vol. 5, No. 2 | 9 July 2007 Cross Ref Approximation of discrete linear systemsInternational Journal of Control, Vol. 17, No. 3 | 27 March 2007 Cross Ref Optimal Observer Techniques for Linear Discrete Time Systems1 1This research was supported in part by the Air Force office of Scientific Research under AFOSR Grant 72–2166.Control and Dynamic Systems | 1 Jan 1973 Cross Ref The product of operators with closed rangeTohoku Mathematical Journal, Vol. 25, No. 3 | 1 Jan 1973 Cross Ref Reduction and transformation of linear discrete-time-varying systems†International Journal of Control, Vol. 16, No. 6 | 1 Dec 1972 Cross Ref A Partial-Step Algorithm for the Nonlinear Estimation ProblemAIAA Journal, Vol. 10, No. 5 | 1 May 1972 Cross Ref The Invariance Properties of a Multivalue n-Port in a Linear EmbeddingIEEE Transactions on Circuit Theory, Vol. 19, No. 2 | 1 Mar 1972 Cross Ref ReferencesRegression and the Moore-Penrose Pseudoinverse | 1 Jan 1972 Cross Ref Das verallgemeinerte Inverse eines linearen Operators in Vektorräumen mit nicht ausgearteter Hermitescher Metrik über einenm kommutativen Körper.Journal für die reine und angewandte Vol. No. | 1 Jan 1972 Cross Ref A method for nonlinear Programming, Vol. 1, No. 1 | 1 Dec Cross Ref Theory of Models of Dynamic Journal, Vol. 9, No. 8 | 1 Aug Cross Ref Generalized and for Singular Operator Analysis and Applications | 1 Jan Cross Ref FOR A WITHOUT Methods in | 1 Jan Cross Ref A Applied to Transactions on Engineering, Vol. No. 1 | 1 Jan Cross Ref An of the Generalized Inverse of a E. and T. 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