Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 283
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Bibi, Nargis Kleerekoper, Anthony Muhammad, Nazeer and Cheetham, Barry 2016. Equation-Method for correcting clipping errors in OFDM signals. SpringerPlus, Vol. 5, Issue. 1,

    Gou, Jin Guo, Wang-Ping Wang, Cheng and Luo, Wei 2016. A multi-strategy improved particle swarm optimization algorithm and its application to identifying uncorrelated multi-source load in the frequency domain. Neural Computing and Applications,

    Low, Ying Min 2016. Importance Sampling Technique for Simulating Time Histories for Efficient Rainflow Fatigue Analysis. Journal of Engineering Mechanics, Vol. 142, Issue. 4, p. 04016006.

    Movahedian, Bashir and Boroomand, Bijan 2016. Inverse identification of time-harmonic loads acting on thin plates using approximated Green’s functions. Inverse Problems in Science and Engineering, Vol. 24, Issue. 8, p. 1475.

    Peng, Zhen-yun Fang, Yang-zhi Xiao, Xian-wei and Du, Dan-dan 2016. New algorithms to compute the nearness symmetric solution of the matrix equation. SpringerPlus, Vol. 5, Issue. 1,

    Shamim, Muhammad Ali Hassan, Muhammad Ahmad, Sameel and Zeeshan, Muhammad 2016. A comparison of Artificial Neural Networks (ANN) and Local Linear Regression (LLR) techniques for predicting monthly reservoir levels. KSCE Journal of Civil Engineering, Vol. 20, Issue. 2, p. 971.

    Sun, Lizhu Zheng, Baodong Bu, Changjiang and Wei, Yimin 2016. Moore–Penrose inverse of tensors via Einstein product. Linear and Multilinear Algebra, Vol. 64, Issue. 4, p. 686.

    Zou, Lei Wang, Zidong and Gao, Huijun 2016. Observer-based <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="" xmlns:xs="" xmlns:xsi="" xmlns="" xmlns:ja="" xmlns:mml="" xmlns:tb="" xmlns:sb="" xmlns:ce="" xmlns:xlink="" xmlns:cals="" xmlns:sa=""><mml:msub><mml:mrow><mml:mi mathvariant="script">H</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msub></mml:math> control of networked systems with stochastic communication protocol: The finite-horizon case. Automatica, Vol. 63, p. 366.

    Cancelliere, Rossella Gai, Mario Gallinari, Patrick and Rubini, Luca 2015. OCReP: An Optimally Conditioned Regularization for pseudoinversion based neural training. Neural Networks, Vol. 71, p. 76.

    Cancelliere, R. Deluca, R. Gai, M. Gallinari, P. and Rubini, L. 2015. An analysis of numerical issues in neural training by pseudoinversion. Computational and Applied Mathematics,

    de Chazal, Philip Tapson, Jonathan and van Schaik, Andre 2015. 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). p. 2165.

    Ding, Derui Wang, Zidong Lam, James and Shen, Bo 2015. Finite-Horizon <inline-formula> <tex-math notation="TeX">${\cal H}_{\infty}$</tex-math></inline-formula> Control for Discrete Time-Varying Systems With Randomly Occurring Nonlinearities and Fading Measurements. IEEE Transactions on Automatic Control, Vol. 60, Issue. 9, p. 2488.

    Erol, H. Ersin and Iftar, Altug 2015. Decentralized Controller Design by Continuous Pole Placement for Incommensurate-Time-Delay Systems∗∗This work is supported in part by the Scientific and Technical Research Council of Turkey (T UB? ITAK) under grant number 112E153 and in part by the Scientific Research Projects Commission of Anadolu University under grant number 1204F071.. IFAC-PapersOnLine, Vol. 48, Issue. 12, p. 257.

    Li, Xun Ni, Yuan-Hua and Zhang, Ji-Feng 2015. Discrete-Time Stochastic Linear-Quadratic Optimal Control with Time-Inconsistency. IFAC-PapersOnLine, Vol. 48, Issue. 28, p. 691.

    Namazi-Rad, Mohammad-Reza and Steel, David 2015. What Level of Statistical Model Should We Use in Small Area Estimation?. Australian & New Zealand Journal of Statistics, Vol. 57, Issue. 2, p. 275.

    Ni, Yuan-Hua Li, Xun and Zhang, Ji-Feng 2015. Finite-Horizon Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control. IFAC-PapersOnLine, Vol. 48, Issue. 28, p. 211.

    Ni, Yuan-Hua Zhang, Ji-Feng and Li, Xun 2015. Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control. IEEE Transactions on Automatic Control, Vol. 60, Issue. 7, p. 1786.

    Sadr, Nadi de Chazal, Philip van Schaik, Andre and Breen, Paul 2015. 2015 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC). p. 7675.

    Safak, S. 2015. The Best Approximate Solution of the Inconsistent Linear System via a Quadratic Penalty Function. Bulletin of the Malaysian Mathematical Sciences Society, Vol. 38, Issue. 2, p. 683.

    Seminara, Lucia Capurro, Marco and Valle, Maurizio 2015. Tactile data processing method for the reconstruction of contact force distributions. Mechatronics, Vol. 27, p. 28.

  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 52, Issue 1
  • January 1956, pp. 17-19

On best approximate solutions of linear matrix equations

  • R. Penrose (a1)
  • DOI:
  • Published online: 24 October 2008

In an earlier paper (4) it was shown how to define for any matrix a unique generalization of the inverse of a non-singular matrix. The purpose of the present note is to give a further application which has relevance to the statistical problem of finding ‘best’ approximate solutions of inconsistent systems of equations by the method of least squares. Some suggestions for computing this generalized inverse are also given.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

(1)A. Bjerhammar Rectangular reciprocal matrices with special reference to geodetic calculations. Bull. géod. int. (1951), pp. 188220.

(3)E. H. Moore Bull. Amer. math. Soc. (2) 26 (1920), 394–5.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *