Abstract
The concept of generalized inverses seems to have been first mentioned in print in 1903 by Fredholm [1] who formulated a pseudoinverse for a linear integral operator which is not invertible in the ordinary sense.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Fredholm, I.: Sur une classe d’equations fonctionnelles. Acta. Math. 27, 365–390 (1903)
Hilbert, D.: Grundzüuge einer algemeinen Theorie der linearen Integralgleichungen, B. G. Teubner, Leipzig and Berlin, 1912, (Reprint of six articles which appeared originally in the Götingen Nachrichten (1904), 49–51; (1904), 213–259; (1905), 307–338; (1906), 157–227; (1906), 439–480; (1910), 355–417)
Hurwitz, W.A.: On the pseudo-resolvent to the kernel of an integral equation. Trans. Am. Math. Soc. 13, 405–418 (1912)
Moore, E.H.: General Analysis. American Philosophical Society, Philadelphia (1935)
Moore, E.H.: On the reciprocal of the general algebraic matrix (abstract). Bull. Am. Math. Soc. 26, 394–395 (1920)
Siegel, C.L.: Uber die analytische Theorie der quadratischen Formen III. Ann. Math. 38, 212–291 (1937)
Siegel, C.L.: Equivalence of quadratic forms. Am. J. Math. 63, 658–680 (1941)
Tseng, Y.Y.: Generalized inverses of unbounded operators between two unitary spaces. Dokl. Akad. Nauk. SSSR. 67, 431–434 (1949)
Tseng, Y.Y.: Properties and classifications of generalized inverses of closed operators. Dokl. Akad. Nauk. SSSR 67, 607–610 (1949)
Tseng, Y.Y.: Virtual solutions and general inversions. Uspehi. Mat. Nauk. 11, 213–215 (1956)
Murray, F.J., von Neumann, J.: On rings of operators. Ann. Math. 37, 116–229 (1936)
Atkinson, F.V.: The normal solvability of linear equations in normed spaces (russian), Mat. Sbornik N.S. 28(70), 3–14 (1951)
Atkinson, F.V.: On relatively regular operators. Acta Sci. Math. Szeged 15, 38–56 (1953)
Bjerhammer, A.: Application of the calculus of matrices to the method of least squares with special reference to geodetic calculations, Kungl. Tekn. H11gsk. Hand. Stockholm. 49, 1–86 (1951)
Bjerhammer, A.: Rectangular reciprocal matrices with special reference to geodetic calculations. Bull. Geodesique 52, 188–220 (1951)
Penrose, R.: A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51, 466–413 (1955)
Rao, C.R.: Analysis of dispersion for multiply classified data with unequal numbers in cells. Sankhyd. 15, 253–280 (1955)
Reid, W.T.: Generalized Green’s matrices for compatible systems of differential equations. Am. J. Math. 53, 443–459 (1931)
Bradley, J.S.: Adjoint quasi-differential operators of Euler type. Pacific J. Math. 16, 213–237 (1966)
Bradley, J.S.: Generalized Green’s matrices for compatible differential systems. Michigan Math. J. 13, 97–108 (1966)
Kammerer, W.J., Nashed, M.Z.: Iterative methods for best approximate solutions of linear integral equations of the first and second kinds. J. Math. Anal. Appl. 40, 547–573 (1972)
Nashed, M.Z.: On moment-discretization and least-squares solutions of linear integral equations of the first kind. J. Math. Anal. Appl. 53, 359–366 (1976)
Kunkel, P., Mehrmann, V.: Generalized inverses of differential-algebraic operators. SIAM J. Matrix Anal. Appl. 17, 426–442 (1996)
Gaarder, N.T., Herman, G.T.: Algorithms for reproducing objects from their x-rays. Comput. Graphics Image Process. 1, 97–106 (1972)
Gordon R., Herman G.T.: Reconstruction of pictures from their projections. Commun. ACM. 14, 759–768 (1971)
Herman, G.T.: Two direct methods for reconstructing pictures from their projections. Comput. Graphics Image Process. 1, 123–143 (1972)
Krishan, S., Prabhu, S.S., Krishnamurthy, E.V.: Probabilistie reinforcement algorithms lot reconsh-uction of pictures from their projections. Int. J. Syst. Sci. 4, 661–670 (1973)
Krishnamurthy, E.V.: Mahadeva, R.T., Subramanian, K., Prabhu, S.S.: Reconstruction of objects from their projections using generalized inverses. Comput. Graphics Image Process. 3, 336–345 (1974)
Rieder, A., Schuster, T.: The approximate inverse in action with an application to computerized tomography SIAM. J. Numer. Anal. 37(6), 1909–1929 (2000)
Laura, A.M., Conde, C.: Generalized inverses and sampling problems. J. Math. Anal. Appl. 398(2), 744–751 (2013)
Li, G., Guo, D.: One-way property proof in public key cryptography based on OHNN. Proc. Eng. 15, 1812–1816 (2011)
Radhakrishin, C.R.: A note on a generalized inverse of a matrix with applications to problems in mathematical statistics. J. R. Stat. Soc. Ser. B. 24, 152–158 (1962)
Radhakrishin, C.R.: Generalized inverse for matrices and its applications in mathematical statistics. Research papers in Statistics. Festschrift for J. Neyman. Wiley, New York (1966)
Radhakrishin, C.R.: Least squares theory using an estimated dispersion matrix and its application to measurement of signals. In: Proceedings of the Fifth Berkeley Symposium on Statistics and Probability, Berkeley and Los Angeles, University of California Press, vol. 1, pp. 355–372 (1967)
Chipman, J.S.: Specification problems in regression analysis, Theory and Application of Generalized Inverses and Matrices, Symposium Proceedings. Texas Technological College. Mathematics Series 4, 114–176 (1968)
Bose, R. C.: Analysis of Variance, unpublished lecture notes, University of North Carolina (1959)
Campbell, S.L., Meyer, C.D.: Generalized Inverse of Linear Transformations, Pitman, London (1979). Dover, New York (1991)
Anderson Jr., W.N., Duffin, R.J., Trapp, G.E.: Matrix operations induced by network connections. SIAM J. Control 13, 446–461 (1975)
Akatsuka, Y., Matsuo, T.: Optimal control of linear discrete systems using the generalized inverse of a matrix, Techn Rept., vol. 13. Institute of Automatic Control, Nagoya Univ., Nagoya, Japan (1965)
Albert, A., Sittler, R.W.: A method for computing least squares estimators that keep up with the data. SIAM J. Control 3, 384–417 (1965)
Balakrishnan, A.V.: An operator theoretic formulation of a class of control problems and a steepest descent method of solution. J. Soc. Ind. Appl. Math. Ser. A. Control 1, 109–127 (1963)
Barnett, S.: Matrices in Control Theory. Van Nostrand Reinhold, London (1971)
Beutler, F.J., Root, W.L.: The operator pseudoinverse in control and systems identification. In: Nashed, M.Z. (ed.) Generalized Inverses and Applications. Academic Press, New York (1976)
Campbell, S.L.: Optimal control of autonomous linear processes with singular matrices in the quadratic cost functional. SIAM J. Control Optim. 14(6), 1092–1106 (1976)
Campbell, S.L: Optimal control of discrete linear processes with quadratic cost. Int. J. Syst. Sci. 9(8), 841–847 (1978)
Campbell, S.L.: Control problem structure and the numerical solution of linear singular systems. Math. Control Signals Syst. 1(1), 73–87 (1988)
Dean, P., Porrill, J.: Pseudo-in J. Math. Anal. Appl.verse control in biological systems: a learning mechanism for fixation stability. Neural Netw. 11, 1205–1218 (1998)
Kalman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5(2), 102–119 (1960)
Kalman, R.E., Ho, Y.C., Narendra, K.S.: Controllability of linear dynamical systems. Contributions to Differential Equations, vol. I, pp. 189–213. Interscience, New York (1963)
Doh-Hyun, K., Jun-Ho, O.: The Moore-Penrose inverse for the classificatory models. Control Eng. Pract. 7(3), 369–373 (1999)
Liu, X., Liu, D.: Recursive computation of generalized inverses with application to optimal state estimation. Control Theory Adv. Tech. 10, 1485–1497 (1995)
Lovass-Nagy, V., Powers, D.L.: Matrix generalized inverses in the handling of control problems containing input derivatives. Int. J. Syst. Sci. 6, 693–696 (1975)
Minamide, N., Nakamura, K.: Minimum error control problem in banach space, Research Report of Automatic Control Lab 16. Nagoya University, Nagoya, Japan (1969)
Wahba, G., Nashed, M.Z.: The approximate solution of a class of constrained control problems. In: Proceedings of the Sixth Hawaii International Conference on Systems Sciences, Hawaii (1973)
Wang, Y.W., Wang, R.J.: Pseudoinverse and two-objective optimal control in Banach spaces. Funct. Approx. Comment. Math. 21, 149–160 (1992)
Doty, K.L., Melchiorri, C., Bonivento, C.: A Theory of Generalized Inverses Applied to Robotics. Int. J. Robotics Res. 12(1), 1–19 (1993)
Schwartz, E.M., Doty, K.L.: Application of the Weighted Generalized-Inverse in Control Optimization and Robotics. Florida Atlantic University, Boca Raton FL (June, Fifth Conf. on Recent Advances in Robotics (1992)
Schwartz, E.M., Doty, K.L.: The Weighted Generalized-Inverse Applied to Mechanism Controllability. University of Florida, Gainesville, FL (April, Sixth Conf. On Recent Advances in Robotics (1993)
Liu, W., Xu, Y., Yao, J., Zhao, Y., Zhao, B.Y., Liu, W.: The weighted Moore-Penrose generalized inverse and the force analysis of overconstrained parallel mechanisms, Multibody System Dynamics, 39, 363–383 (2017)
Lasky, T.A., Ravani, B.: Sensor-based path planning and motion control for a robotic system for roadway crack sealing. IEEE Trans. Control Syst. Technol. 8, 609–622 (2000)
Tucker, M., Perreira, N.D.: Generalized inverses for robotics manipulators. Mech. Mach. Theory 22(6), 507–514 (1987)
Lenarčić, J., Husty, M.: Latest Advances in Robot Kinematic. Springer (2012)
Angeles, J.: The application of dual algebra to kinematic analysis. In: Angeles, J., Zakhariev, E. (eds.) Computational Methods in Mechanical Systems, pp. 3–31. Springer, Heidelberg (1998)
Angeles, J.: The Dual Generalized Inverses and Their Applications in Kinematic Synthesis. In: Lenarcic, J., Husty, M. (eds.) Latest Advances in Robot Kinematics, pp. 1–10. Springer, Netherlands (2012)
Drazin, M.P.: Pseudoinverse in associative rings and semigroups. Am. Math. Monthly 65, 506–514 (1958)
Hartwig, R.E.: Schur’s theorem and the Drazin inverse. Pacific J. Math. 78, 133–138 (1978)
Ben-Isreal, A., Greville. T.N.E.: Generalized Inverse: Theory and Applications, 2nd edn. Springer, New York (2003)
Campbell, S.L., Meyer, C.D., Rose, N.J.: Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31, 411–425 (1976)
Nashed, M.Z., Zhao, Y.: The Drazin inverse for singular evolution equations and partial differential equations. World Sci. Ser. Appl. Anal. 1, 441–456 (1992)
Simeon, B., Fuhrer, C., Rentrop, P.: The Drazin inverse in multibody system dynamics. Numer. Math. 64, 521–539 (1993)
Caradus, S.R.: Generalized inverses and operator theory, Queen’s Papers in Pure and Appl. Math. 50, Queen’s University, Kingston, Ontario (1978)
King, C.F.: A note on Drazin inverses. Pacific J. Math. 70, 383–390 (1977)
Lay, D.C.: Spectral properties of generalized inverses of linear operators. SIAM J. Appl. Math. 29, 103–109 (1975)
Marek, I., Zitny, K.: Matrix Analysis for Applied Sciences, 2, Teubner-Texte zur Mathematik, vol. 84. Teubner, Leipzig (1986)
Drazin, M.P.: Extremal definitions of generalized inverses. Linear Algebra Appl. 165, 185–196 (1992)
Heuser, H.G.: Functional Analysis. Wiley, New York (1982)
Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis, 2nd edn. Wiley, New York (1980)
Hunter, J.: Generalized Inverses and their application to applied probability problems. Linear Algebra Appl. 45, 157–198 (1982)
Hunter, J.: Generalized inverses of Markovian kernels in terms of properties of the Markov chain. Linear Algebra Appl. 447, 38–55 (2014)
Meyer, C.D.: The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17(3), 443–464 (1975)
Ben-Israel, A., Charnes, A.: Generalized inverses and the Bott-Duffin network analysis. J. Math. Anal. Appl. 7, 428–435 (1963)
Ben-Israel, A.: Linear equations and inequalities on finite dimensional, real or complex vector spaces: a unified theory. J. Math. Anal. Appl. 27, 367–389 (1969)
Ben-Israel, A.: A note on partitioned matrices and equations. SIAM Rev. 11, 247–250 (1969)
Berman, A., Plemmons, R.J.: Cones and iterative methods for best least-squares solutions of linear systems. SIAM J. Numer. Anal. 11, 145–154 (1974)
Cline, R.E.: Representations for the generalized inverse of a partitioned matrix. SIAM J. Appl. Math. 12, 588–600 (1964)
Anderson, W.N. Jr., Trapp, G.E.: Shorted operators II. SIAM J. Appl. Math. 28, 60–71 (1975)
Bart, H., Kaashoek, M.A., Lay, D.C.: Relative inverses of meromorphic operator functions and associated holomorphic projection functions. Math. Ann. 218(3), 199–210 (1975)
Ben-Israel, A.: A note on an iterative method for generalized inversion of matrices. Math. Comput. 20, 439–440 (1966)
Campbell, S.L.: The Drazin inverse of an infinite matrix. SIAM J. Appl. Math. 31, 492–503 (1976)
Campbell, S.L.: Linear systems of differential equations with singular coefficients. SIAM J. Math. Anal. 8, 1057–1066 (1977)
Campbell, S.L.: On the limit of a product of matrix exponentials. Linear Multilinear Algebra 6, 55–59 (1978)
Campbell, S.L.: Singular perturbation of autonomous linear systems II. J. Differ. Eqn. 29, 362–373 (1978)
Campbell, S.L.: Limit behavior of solutions of singular difference equations. Linear Algebra Appl. 23, 167–178 (1979)
Faddeev, D.K., Faddeeva, V.N.: Computational Methods of Linear Algebra, (translated by Robert C. Williams). W.H. Freeman and Co., San Francisco (1963)
Gantmacher, F.R.: The Theory of Matrices, vol. II. Chelsea Publishing Company, New York (1960)
Cederbaum, I.: On equivalence of resistive n-port networks. IEEE Trans. Circuit Theory CT-12, 338–344 (1965)
Cederbaum, I., Lempel, A.: Parallel connection of n-port networks, IEEE Trans. Circuit Theory CT-14, 274–279 (1967)
Ben-Israel, A.: An iterative method for computing the generalized inverse of an arbitrary matrix. Math. Comput. 19, 452–455 (1965)
Ben-Israel, A.: On error bounds for generalized inverses. SIAM J. Numer. Anal. 3, 585–592 (1966)
Campbell, S.L.: Differentiation of the Drazin inverse. SIAM J. Appl. Math. 30, 703–707 (1976)
Churchill, R.V.: Operational Mathematics. McGraw-Hill, New York (1958)
Golub, G., Kahan, W.: Calculating the singular values and pseudoinverse of a matrix. SIAM J. Numer. Anal. Series B 2, 205–224 (1965)
Golub, G.H., Pereyra, V.: The differentiation of pseudo-inverses and non-linear least squares problems whose variables separate. SIAM J. Numer. Anal. 10, 413–432 (1973)
Golub, G.H., Wilkinson, J.H.: Ill conditioned eigensystems and the computation of the Jordan canonical form. SIAM Rev. 18(4), 578–619 (1976)
Golub, G.H., Reinsch, C.: Singular value decomposition and least squares solutions. Numer. Math. 14, 403–420 (1970)
Greville, T.N.E.: Spectral generalized inverses of square matrices, MRC Tech. Sum. Rep. 823, Mathematics Research Center, University of Wisconsin, Madison (1967)
Kirkland, S.J., Neumann, M.: Group Inverses of M-Matrices and their Applications. CRC Press (2012)
Rao, C.R.: A note on a generalized inverse of a matrix with applications to problems in mathematical statistics, J. R. Stat. Soc. Ser. B Stat. Methodol. 24(1), 152–158 (1962)
Riaza, R.: Differential-Algebriac Systems. Analytical Aspects and Circuit Applications, World Scientifics, River Edge, NJ (2008)
Koliha, J.J.: A generalized Drazin inverse. Glasgow Math. J. 38, 367–381 (1996)
Harte, R.E.: Spectral projections. Irish Math. Soc. Newslett. 11, 10–15 (1984)
Harte, R.E.: Invertibility and Singularity for Bounded Linear Operators. Marcel Dekker, New York (1988)
Harte, R.E.: On quasinilpotents in rings. PanAm. Math. J. 1, 10–16 (1991)
Han, J.K., Lee, H.Y., Lee, W.Y.: Invertible completions of \(2 \times 2\) upper triangular operator matrices. Proc. Am. Math. Soc. 128, 119–123 (1999)
Koliha, J.J., Tran, T.D.: Semistable operators and singularly perturbed differential equations. J. Math. Anal. Appl. 231, 446–458 (1999)
Koliha, J.J., Tran, T.D.: The Drazin inverse for closed linear operators. J. Oper. Theory 46(2), 323–336 (2001)
Campbell, S.L.: The Drazin inverse and systems of second order linear differential equations. Linear Multilinear Algebra 14, 195–198 (1983)
Cui, X., Wei, Y., Zhang, N.: Quotient convergence and multi-splitting methods for solving singular linear equations. Calcolo 44, 21–31 (2007)
Hooker, J.W., Langenhop, C.E.: On regular systems of linear differential equations with constant coefficients. Rocky Mountain J. Math. 12(4), 591–614 (1982)
Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebriac Equations. SIAM, Philadelphia (1998)
Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Classics in Appl. Math. 14, SIAM, Philadelphia (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Cvetković Ilić, D.S., Wei, Y. (2017). Definitions and Motivations. In: Algebraic Properties of Generalized Inverses. Developments in Mathematics, vol 52. Springer, Singapore. https://doi.org/10.1007/978-981-10-6349-7_1
Download citation
DOI: https://doi.org/10.1007/978-981-10-6349-7_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-6348-0
Online ISBN: 978-981-10-6349-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)