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Generalized inverses of matrices: a perspective of the work of Penrose

Published online by Cambridge University Press:  24 October 2008

A. Ben-Israel
A. Ben-Israel
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A.
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Extract

As a mathematical area generalized inversion1 was inaugurated in 1955 by R. Penrose [128]. Since then there have appeared about 2000 articles and 15 books2 on generalized inverses of matrices and linear operators. See the annotated bibliography by Nashed and Rall [121] for the period up to 1976.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 3 , November 1986 , pp. 407 - 425
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Afriat, S. N.. Orthogonal and oblique projectors and the characteristics of pairs of vector spaces. Proc. Cambridge Philos. Soc. 53 (1957), 800–816.CrossRefGoogle Scholar
[2]Albert, A.. Regression and the Moore-Penrose Pseudoinverse (Academic Press, 1972).Google Scholar
[3]Albert, A.. Statistical applications of the pseudo inverse, pp. 525–548 in [118].CrossRefGoogle Scholar
[4]Anderson, W. N. Jr and Duffin, R. J.. Series and parallel addition of matrices. J. Math. Anal. Appl. 26 (1969), 576–594.CrossRefGoogle Scholar
[5]Anselone, P. M. and Nashed, M. Z.. Perturbation of outer inverses. In Approximation Theory III, ed Cheney, E. W. (Academic Press, 1980), 163–169.Google Scholar
[6]Atkinson, F. V.. On relatively regular operators. Acta Sci. Math. (Szeged) 15 (1953), 38–56.Google Scholar
[7]Autonne, L.. Sur les matrices hypohermitiennes et sur les matrices unitaires. Ann. Univ. Lyon Sect. A 38 (1915), 1–77.Google Scholar
[8]Ben-Israel, A.. A Newton-Raphson method for the solution of systems of equations. J. Math. Anal. Appl. 15 (1966), 243–252.CrossRefGoogle Scholar
[9]Ben-Israel, A.. Applications of generalized inverses in nonlinear analysis, pp. 183–202 in [32].Google Scholar
[10]Ben-Israel, A.. Applications of generalized inverses to programming, games and networks, pp. 495–523 in [118].CrossRefGoogle Scholar
[11]Ben-Israel, A.. Generalized inverses of matrices and their applications, pp. 154–186 in [67].CrossRefGoogle Scholar
[12]Ben-Israel, A.. A Cramer rule for least-norm solutions of consistent linear equations. Linear Algebra Appl. 43 (1982), 223–226.Google Scholar
[13]Ben-Israel, A. and Charnes, A.. Contributions to the theory of generalized inverses. J. Soc. Indust. Appl. Math. 11 (1963), 667–699.Google Scholar
[14]Ben-Israel, A. and Charnes, A.. Generalized inverses and the Bott-Duffin network analysis. J. Math. Anal. Appl. 7 (1963), 428–435.Google Scholar
Erratum, Generalized inverses and the Bott-Duffin network analysis. J. Math. Anal. Appl. 18 (1967), 393.Google Scholar
[15]Ben-Israel, A. and Charnes, A.. An explicit solution of a special class of linear programming problems. Oper. Res. 16 (1968), 1167–1175.Google Scholar
[16]Ben-Israel, A. and Greville, T. N. E.. Generalized Inverses: Theory and Applications (J. Wiley, 1974). (Reprinted by R. E. Krieger 1980.)Google Scholar
[17]Ben-Israel, A. and Greville, T. N. E.. Some topics in generalized inverses of matrices, pp. 125–147. In [118].Google Scholar
[18]Bergmann, P. G., Penfield, R., Schiller, R. and Zatzkis, H.. The Hamiltonian of the general theory of relativity with electromagnetic field. Phys. Rev. 80 (1950), 81–88.Google Scholar
[19]Berman, A. and Plemmons, R. J.. Monotonicity and the generalized inverse. SIAM J. Appl. Math. 22 (1972), 155–161.Google Scholar
[20]Bernkopf, M.. The development of function spaces with particular reference to their origins in integral equation theory. Arch. Hist. Exact Sci. 3 (1966), 1–96.CrossRefGoogle Scholar
[21]Bernkopf, M.. A history of infinite matrices: A study of denumerably infinite linear systems as the first step in the history of operators defined on function spaces. Arch. Hist. Exact Sci. 4 (1968), 308–358.Google Scholar
[22]Beutler, F. J.. The operator theory of the pseudoinverse I. Bounded operators. J. Math. Anal. Appl. 10 (1965), 451–470.CrossRefGoogle Scholar
II. Unbounded operators, The operator theory of the pseudoinverse I. Bounded operators. J. Math. Anal. Appl. 10 (1965), 471–493.Google Scholar
[23]Beutler, F. J. and Root, W. L.. The operator pseudoinverse in control and system identification, pp. 397–494 in [118].CrossRefGoogle Scholar
[24]Bjerhammar, A.. Rectangular reciprocal matrices, with special reference to geodetic calculations. Bull Géodésique 52 (1951), 188–220.Google Scholar
[25]Bjerhammar, A.. A New Matrix Algebra (Swedish). Svensk Lantmäteritidskrift häfte 5–6. Bulletin No. 2, Division of Geology, Royal Institute of Technology, Stockholm (1955), 311–388.Google Scholar
[26]Bjerhammar, A.. A Theory of Errors and Generalized Inverse Matrices (Elsevier, 1973).Google Scholar
[27]Björk, A. and Golub, G. H.. Numerical methods for computing angles between linear subspaces. Math. Comp. 27 (1973), 579–594.CrossRefGoogle Scholar
[28]Bliss, G. A.. Eliakim Hastings Moore. Bull. Amer. Math. Soc. 39 (1933), 831–838.Google Scholar
[29]Bliss, G. A.. The scientific work of Eliakim Hastings Moore. Bull. Amer. Math. Soc. 40 (1934), 501–514.Google Scholar
[30]Boggs, P. T.. The convergence of the Ben-Israel iteration for nonlinear least squares problems. Math. Comp. 30 (1976), 512–522.Google Scholar
[31]Bott, R. and Duffin, R. J.. On the algebra of networks. Trans. Amer. Math. Soc. 74 (1953), 99–109.CrossRefGoogle Scholar
[32]Boullion, T. L. and Odell, P. L. (ed), Proceedings of the Symposium on Theory and Applications of Generalized Inverses of Matrices (Texas Tech. Press, 1968).Google Scholar
[33]Boullion, T. L. and Odell, P. L.. Generalized Inverse Matrices (Wiley, 1971).Google Scholar
[34]Bowman, V. J. and Burdet, C. A.. On the general solution to systems of mixed-integer linear equations. SIAM J. Appl. Math. 26 (1974), 120–125.Google Scholar
[35]Cajori, F.. A History of Mathematical Notation, Volume II. Notations mainly in Higher Mathematics (Open Court, 1929).Google Scholar
[36]Campbell, S. L.. Differentiation of the Drazin inverse. SIAM J. Appl. Math. 30 (1976), 703–707.Google Scholar
[37]Campbell, S. L.. Optimal control of autonomous linear processes with singular matrices in the quadratic cost functional. SIAM J. Control Optim. 14 (1976), 1092–1106.Google Scholar
[38]Campbell, S. L.. Linear systems of differential equations with singular coefficients. SIAM J. Math. Anal. 8 (1977), 1057–1066.Google Scholar
[39]Campbell, S. L.. Optimal control of discrete linear processes with quadratic cost. Internat. J. Systems Sci. 9 (1978), 841–847.Google Scholar
[40]Campbell, S. L. (ed), Recent Applications of Generalized Inverses (Pitman, 1982).Google Scholar
[41]Campbell, S. L.. The Drazin inverse of an operator, pp. 250–260 in [40].Google Scholar
[42]Campbell, S. L. and Meyer, C. D. Jr. Generalized Inverses of Linear Transformations (Pitman, 1979).Google Scholar
[43]Campbell, S. L., Meyer, C. D. Jr and Rose, N. J.. Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31 (1976), 411–425.Google Scholar
[44]Chang, E.. The generalized inverse and interpolation theory, pp. 196–219 in [40].Google Scholar
[45]Charnes, A. and Cooper, W. W.. Structural sensitivity analysis in linear programming and an exact product form left inverse. Naval Res. Logist. Quart. 15 (1968), 517–522.Google Scholar
[46]Chernoff, H.. Locally optimal designs for estimating parameters. Ann. Math. Statist. 24 (1953), 586–602.Google Scholar
[47]Chipman, J. S.. On least squares with insufficient observations. J. Amer. Statist. Assoc. 54 (1964), 1078–1111.Google Scholar
[48]Chipman, J. S.. Estimation and aggregation in econometrics: An application of the theory of generalized inverses, pp. 549–769 in [118].CrossRefGoogle Scholar
[49]Cline, R. E.. Inverses of rank invariant powers of a matrix. SIAM J. Numer. Anal. 5 (1968), 182–197.Google Scholar
[50]Cline, R. E.. Note on an extension of the Moore-Penrose inverse. Linear Algebra Appl. 30 (1981), 19–24.Google Scholar
[51]Cline, R. E. and Greville, T. N. E.. A Drazin inverse for rectangular matrices. Linear Algebra Appl. 29 (1980), 53–62.Google Scholar
[52]Cline, R. E. and Pyle, L. D.. The generalized inverse in linear programming. An intersection projection method and the solution of a class of structured linear programming problems. SIAM J. Appl. Math. 5 (1972), 329–338.Google Scholar
[53]Coray, C. S.. Parameter estimation using minimum norm differential approximation. J. Math. Anal. Appl. 64 (1978), 159–165.Google Scholar
[54]Davis, D. L. and Robinson, D. W.. Generalized inverses of morphisms. Linear Algebra Appl. 5 (1972), 329–338.Google Scholar
[55]Deutsch, E.. Semi-inverses, reflexive semi-inverses, and pseudoinverses of an arbitrary linear transformation. Linear Algebra Appl. 4 (1971), 313–322.Google Scholar
[56]Drazin, M. P.. Pseudoinverses in associative rings and semigroups. Amer. Math. Monthly 65 (1958), 506–524.Google Scholar
[57]Drazin, M. P.. Differentiation of generalized inverses, pp. 138–144 in [40].Google Scholar
[58]Drygas, H.. The Coordinate-Free Approach to Gauss-Markov Estimation (Springer-Verlag, 1970).Google Scholar
[59]Duffin, R. J. and Morley, T. D.. Inequalities induced by network connections, pp. 27–49 in [40].Google Scholar
[60]Eckart, C. and Young, G.. The approximation of one matrix by another of lower rank. Psychometrika 1 (1936), 211–218.CrossRefGoogle Scholar
[61]Eckart, C. and Young, G.. A principal axis transformation for non-Hermitian matrices. Bull. Amer. Math. Soc. 45 (1939), 118–121.Google Scholar
[62]Engl, H. W. and Nashed, M. Z.. New characterizations of generalized inverses of linear operators. J. Math. Anal. Appl. 82 (1981), 566–586.Google Scholar
[63]Englefield, M. J.. The commuting inverse of a square matrix. Proc. Cambridge Philos. Soc. 62 (1966), 667–671.Google Scholar
[64]Erdelsky, P. J.. Projections in a Normed Linear Spaces and a Generalization of the Pseudo-Inverse, doctoral dissertation in mathematics (California Institute of Technology, 1969).Google Scholar
[65]Erdelyi, I.. A generalized inverse for arbitrary operators between Hilbert spaces. Proc. Cambridge Philos. Soc. 71 (1972), 43–50.Google Scholar
[66]Erdelyi, I. and Ben-Israel, A.. Extremal solutions of linear equations and generalized inversion between Hilbert spaces. J. Math. Anal. Appl. 39 (1972), 298–313.Google Scholar
[67]Fiacco, A. V. and Kortanek, K. O. (ed). Extremal Methods and Systems Analysis, Lecture Notes in Economics and Mathematical Systems, Vol. 174 (Springer-Verlag, 1980).Google Scholar
[68]Fletcher, R.. Generalized inverse methods for the best least-squares solution of systems of nonlinear equations. Comput. J. 10 (1968), 392–399.Google Scholar
[69]Foulis, D. J.. Relative inverses in Baer*-semigroups. Michigan Math. J. 10 (1963), 65–84.Google Scholar
[70]Fredholm, I.. Sur une classe d'équations fonctionnelles. Acta Math. 27 (1903), 365–390.Google Scholar
[71]Funderlic, R. E. and Meyer, C. D. Jr. Sensitivity of the stationary distribution vector for an ergodic Markov chain. Linear Algebra Appl. 76 (1986), 1–17.Google Scholar
[72]Goldman, A. J. and Zelen, M.. Weak generalized inverses and minimum variance linear unbiased estimation. J. Res. Nat. Bur. Standards Sect. B. 68B (1964), 151–172.Google Scholar
[73]Golub, G. H.. Least squares, singular values and matrix approximations. Apl. Mat. 13 (1968), 44–51.Google Scholar
[74]Golub, G. H. and Kahan, W.. Calculating the singular values and pseudo-inverse of a matrix. SIAM J. Numer. Anal. 2 (1965), 205–224.Google Scholar
[75]Golub, G. H. and Pereyra, V.. Differentiation of pseudoinverses, separable nonlinear least squares problems and other tales, pp. 303–324. In [118].Google Scholar
[76]Golub, G. H. and Reinsch, C.. Singular value decomposition and least squares solutions. Numer. Math. 14 (1970), 403–420.Google Scholar
[77]Greville, T. N. E.. Some new generalized inverses with spectral properties, pp. 26–46 in [32].Google Scholar
[78]Greville, T. N. E.. The Souriau-Frame algorithm and the Drazin pseudoinverse. Linear Algebra Appl. 6 (1973), 205–208.Google Scholar
[79]Groetsch, C. W.. Generalized Inverses of Linear Operators. Representation and Approximation (Marcel Dekker, 1977).Google Scholar
[80]Hartwig, R. E.. Drazin inverses and canonical forms in Mn(Z/h). Linear Algebra Appl. 37 (1981), 205–233.Google Scholar
[81]Hartwig, R. E. and Luh, J.. On finite regular rings. Pacific J. Math. 69 (1977), 73–95.Google Scholar
[82]Hawkins, J. B. and Ben-Israel, A.. On generalized matrix functions. Linear and Multilinear Algebra 1 (1973), 163–171.Google Scholar
[83]Hestenes, M. R.. A ternary algebra with applications to matrices and linear transformations. Arch. Rational Mech. Anal. 11 (1965), 138–194.Google Scholar
[84]Hestenes, M. R.. A role of the pseudo-inverse in analysis, pp. 175–192 in [118].Google Scholar
[85]Hilbert, D.. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (Teubner 1912). (Reprints of six articles which appeared originally in the Göttingen Nachrichten (1904), 49–51; (1904), 213–259; (1905), 307–338; (1906), 157–227; (1906), 439–480; (1910), 355–417.Google Scholar
[86]Hoerl, A. E.. Application of ridge analysis to regression problems. Chem. Engng. Prog. 58 (1962), 54–59.Google Scholar
[87]Hurt, M. F. and Waid, C.. A generalized inverse which gives all the integral solutions to a system of linear equations. SIAM J. Appl. Math. 19 (1970), 547–550.Google Scholar
[88]Hurwitz, W. A.. On the pseudo-resolvent to the kernel of an integral equation. Trans. Amer. Math. Soc. 13 (1912), 405–418.Google Scholar
[89]Kalman, R. E.. A new approach to linear filtering and prediction problems. Trans. ASME Ser. D: J. Basic Engrg. 82 (1960), 35–45.Google Scholar
[90]Kalman, R. E.. New results in linear filtering and prediction theory. Trans. ASME Ser. D: J. Basic Engrg. 83 (1961), 95–107.Google Scholar
[91]Kalman, R. E.. Algebraic aspects of the generalized inverse of a rectangular matrix, pp. 111–124 in [118].Google Scholar
[92]Kishi, F. H.. On line computer techniques and their application to re-entry aerospace vehicle control. In Advances in Control Systems Theory and Applications (ed. Leondes, C. T.) (Academic Press, 1964), pp. 245–257.Google Scholar
[93]Lam, Y.-F. and McPherson, J. D.. An iterative procedure for solving nonlinear equations. J. Math. Anal. Appl. 58 (1977), 578–604.Google Scholar
[94]Landesman, E. M.. Hilbert-space methods in elliptic partial differential equations. Pacific J. Math. 21 (1967), 113–131.Google Scholar
[95]Langenhop, C. E.. On generalized inverses of matrices. SIAM J. Appl. Math. 15 (1967), 1239–1246.Google Scholar
[96]Laurent, P.-J.. Approximation et Optimisation (Hermann, 1972).Google Scholar
[97]Lewis, F.. A generalized inverse solution to the discrete time singular Riccati equation. IEEE Trans. Automat. Control AG-26 (1981), 395–398.Google Scholar
[98]Loud, W. S.. Generalized inverses and generalized Green's functions. SIAM J. Appl. Math. 14 (1966), 342–369.Google Scholar
[99]Loud, W. S.. Some examples of generalized Green's functions and generalized Green's matrices. SIAM Rev. 12 (1970), 194–210.Google Scholar
[100]Loud, W. S.. A bifurcation application of the generalized inverse of a linear differential operator. SIAM J. Math. Anal. 3 (1980), 545–558.Google Scholar
[101]Lovass-Nagy, V., Miller, R. J. and Powers, D. L.. On output control in the servomechanism sense. Internat. J. Control 24 (1976), 435–440.Google Scholar
[102]Lovass-Nagy, V., Miller, R. J. and Powers, D. L.. On the application of matrix generalized inverses to the construction of inverse systems. Internat. J. Control 24 (1976), 733–739.Google Scholar
[103]Lovass-Nagy, V. and Powers, D. L.. On rectangular systems of differential equations and their application to circuit theory. J. Franklin Inst. 299 (1975), 399–407.Google Scholar
[104]MacDuffee, C. C.. The Theory of Matrices (Chelsea, 1956).Google Scholar
[105]McCoy, N. H.. Generalized regular rings. Bull. Amer. Math. Soc. 45 (1939), 175–178.Google Scholar
[106]Meyer, C. D. Jr. The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17 (1975), 443–464.Google Scholar
[107]Meyer, C. D. Jr. Analysis of finite Markov chains by group inversion techniques, pp. 50–81 in [40].Google Scholar
[108]Milne, R. D.. An oblique matrix pseudoinverse. SIAM J. Appl. Math. 16 (1968), 931–944.Google Scholar
[109]Minamide, N. and Nakamura, K.. A restricted pseudoinverse and its application to constrained minima. SIAM J. Appl. Math. 19 (1970), 167–177.Google Scholar
[110]Minamide, N. and Nakamura, K.. A minimum cost control problem in Banach space. J. Math. Anal. Appl. 35 (1971), 222–234.Google Scholar
[111]Moore, E. H.. On the reciprocal of the general algebraic matrix. Bull. Amer. Math. Soc. 26 (1920), 394–395.Google Scholar
[112]Moore, E. H.. General Analysis, Part I. The Algebra of Matrices (compiled and edited by Barnard, R. W.) (The Amer. Philos. Soc. 1935).Google Scholar
[113]Moore, E. H.. General Analysis, Part II. The Fundamental Notions of General Analysis (compiled and edited by Barnard, R. W.) (The Amer. Philos. Soc., 1939).Google Scholar
[114]Morley, T. D.. A Gauss-Markov theorem for infinite dimensional regression models with possibly singular covariance. SIAM J. Appl. Math. 37 (1979), 257–260.Google Scholar
[115]Morley, T. D.. Parallel summation, Maxwell's principle and the infimum of projections.Google Scholar
[116]Munn, W. D.. Pseudoinverses in semigroups. Proc. Cambridge Philos. Soc. 57 (1961), 247–250.Google Scholar
[117]Munn, W. D. and Penrose, R.. A note on inverse semigroups. Proc. Cambridge Philos. Soc. 51 (1955), 396–399.Google Scholar
[118]Nashed, M. Z. (ed). Generalized Inverses and Applications (Academic Press, 1976).Google Scholar
[119]Nashed, M. Z.. Aspects of generalized inverses in analysis and regularization, pp. 193–244 in [118].Google Scholar
[120]Nashed, M. Z.. Perturbations and approximations for generalized inverses and linear operator equations, pp. 325–396 in [118].Google Scholar
[121]Nashed, M. Z. and Rall, L. B.. Annotated bibliography on generalized inverses and applications, pp. 771–1041 in [118].Google Scholar
[122]Nashed, M. Z. and Vortuba, G. F.. A unified approach to generalized inverses of linear operators, pp. 1–109 in [118].Google Scholar
[123]von Neumann, J.. On regular rings. Proc. Nat. Acad. Sci. U.S.A. 22 (1936), 707–713.Google Scholar
[124]Newman, T. G. and Odell, P. L.. On the concept of a p-q generalized inverse of a matrix. SIAM J. Appl. Math. 17 (1969), 520–525.Google Scholar
[125]Noble, B.. Methods for computing the Moore-Penrose generalized inverse and related matters, pp. 245–301 in [118].Google Scholar
[126]Noble, B. and Daniel, J. W.. Applied Linear Algebra, 2nd ed (Prentice-Hall, 1977).Google Scholar
[127]Pearl, M. H.. Generalized inverses of matrices with entries taken from an arbitrary field. Linear Algebra Appl. 1 (1968), 571–587.Google Scholar
[128]Penrose, R.. A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51 (1955), 406–413.Google Scholar
[129]Penrose, R.. On best approximate solution of linear matrix equations. Proc. Cambridge Philos. Soc. 52 (1956), 17–19.Google Scholar
[130]Peters, G. and Wilkinson, J. H.. The least squares problem and pseudoinverses. Comput. J. 13 (1970), 309–316.Google Scholar
[131]Plemmons, R. J.. Generalized inverses of Boolean relation matrices. SIAM J. Appl. Math. 20 (1971), 426–433.Google Scholar
[132]Plemmons, R. J. and Cline, R. E.. The generalized inverse of a nonnegative matrix. Proc. Cambridge Philos. Soc. 31 (1972), 46–50.Google Scholar
[133]Porter, W. A.. A basic optimization problem in linear systems. Math. Systems Theory 5 (1971), 20–44.Google Scholar
[134]Porter, W. A. and Williams, J. P.. A note on the minimum effort control problem. J. Math. Anal. Appl. 13 (1966), 251–264.Google Scholar
[135]Porter, W. A. and Williams, J. P.. Extensions of the minimum effort control problem. J. Math. Anal. Appl. 13 (1966), 536–549.Google Scholar
[136]Pringle, R. J. and Rayner, A. A.. Generalized Inverse Matrices with Applications to Statistics (Griffin, 1971).Google Scholar
[137]Puystijens, R. and Robinson, D. W.. The Moore-Penrose inverses of a morphism with factorization. Linear Algebra Appl. 40 (1981), 129–144.Google Scholar
[138]Pyle, L. D.. The generalized inverse in linear programming. Basic structure. SIAM J. Appl. Math. 22 (1972), 335–355.Google Scholar
[139]Pyle, L. D.. The weighted generalized inverse in nonlinear programming - active set selection using a variable-metric generalization of the Simplex algorithm, pp. 197–230 in [67].Google Scholar
[140]Rado, R.. Note on generalized inverses of matrices. Proc. Cambridge Philos. Soc. 52 (1956), 600–601.Google Scholar
[141]Rall, L. B.. The Fredholm pseudoinverse - an analytic episode in the history of generalized inverses, pp. 149–174 in [118].Google Scholar
[142]Rao, C. R.. A note on a generalised inverse of a matrix with applications to problems in mathematical statistics. J. Roy. Statist. Soc. Ser. B. 24 (1962), 152–158.Google Scholar
[143]Rao, C. R.. Calculus of generalized inverses of matrices, part I: General theory. Sankhyä Ser. A. 29 (1967), 317–342.Google Scholar
[144]Rao, C. R. and Mitra, S. K.. Generalized Inverse of Matrices and its Applications (Wiley, 1971).Google Scholar
[145]Rao, C. R. and Mitra, S. K.. Theory and application of constrained inverse of matrices. SIAM J. Appl. Math. 24 (1973), 473–488.Google Scholar
[146]Reid, W. T.. Generalized inverses of differential and integral operators, pp. 1–25. In [32].Google Scholar
[147]Robert, P.. On the group-inverse of a linear transformation. J. Math. Anal. Appl. 22 (1968), 658–669.Google Scholar
[148]Robinson, D. W.. On the generalized inverse of an arbitrary linear transformation. Amer. Math. Monthly 69 (1962), 412–416.Google Scholar
[149]Searle, S. R.. Matrix Algebra useful for Statistics (Wiley, 1982).Google Scholar
[150]Stakgold, I.. Branching of solutions of nonlinear equations. SIAM Rev. 13 (1971), 289–332.Google Scholar
[151]Stewart, G. W.. On the perturbation of pseudoinverses, projections and linear least squares problems. SIAM Rev. 19 (1977), 634–662.Google Scholar
[152]Strang, G.. Linear Algebra and its Applications (Academic Press 1976).Google Scholar
[153]Sutherland, W. R. S., Wolkowicz, H. and Zeidan, V. M.. An explicit linear solution for the quadratic dynamic programming problem. (To appear.)Google Scholar
[154]Tam, B.-S.. Generalized inverses of cone preserving maps. Linear Algebra Appl. 40 (1980), 189–202.Google Scholar
[155]Tam, B.-S.. A geometric treatment of generalized inverses and semigroups of nonnegative matrices. Linear Algebra Appl. 41 (1981), 225–272.Google Scholar
[156]Tanabe, K.. Differential geometric approach to extended GRG methods with enforced feasibility in nonlinear analysis: Global analysis, pp. 100–137 in [40].Google Scholar
[157]Tseng, Y. Y.. The Characteristic Value Problem of Hermitian Functional Operators in a Non-Hilbert Space, doctoral dissertation in Mathematics, University of Chicago, 1933 (published by the University of Chicago Libraries, 1936).Google Scholar
[158]Tseng, Y. Y.. Sur les solutions des équations opératrices fonctionnelles entre les espaces unitaires. Solutions extrémales. Solutions virtuelles. C. R. Acad. Sci. Paris 228 (1949), 640–641.Google Scholar
[159]Tseng, Y. Y.. Generalized inverses of unbounded operators between two unitary space. Dokl. Akad. Nauk SSSR (N.S.) 67 (1949), 431–434.Google Scholar
[160]Tseng, Y. Y.. Properties and classification of generalized inverses of closed operators. Dokl. Akad. Nauk SSSR (N.S.) 67 (1949), 607–610.Google Scholar
[161]Tseng, Y. Y.. Virtual solutions and general inversions. Uspehi Mat. Nauk (N.S.) 11 (1956), 213–215.Google Scholar
[162]Verghese, G. C.. A ‘Cramer rule’ for the least-norm, least-squared-error solution of inconsistent linear equations. Linear Algebra Appl. 48 (1982), 315–316.Google Scholar
[163]Wedderburn, J. H. M.. Lectures on Matrices, Amer. Math. Soc. Colloq. Publ. Vol. XVII (Providence, R.I., 1934).Google Scholar
[164]Wedin, P.-Å.. Perturbation theory for pseudo-inverses. BIT 13 (1973), 217–232.Google Scholar
[165]Wilkinson, J. H.. Note on the practical significance of the Drazin inverse, pp. 82–99 in [40].Google Scholar
[166]Wolkowicz, H. and Zlobec, S.. Calculating the best approximate solution of an operator equation. Math. Comput. 32 (1978), 1183–1213.Google Scholar
[167]Wong, E. T.. Involutory functions and Moore-Penrose inverses of matrices in an arbitrary field. Linear Algebra Appl. 48 (1982), 283–291.Google Scholar
[168]Wong, Y. K.. Spaces associated with Non-Modular Matrices with Application to Reciprocals, doctoral dissertation in Mathematics, University of Chicago, 1931.Google Scholar
[169]Zarantonello, E. H.. Differentioids. Adv. in Math. 2 (1968), 187–306.Google Scholar
[170]Zlobec, S.. On computing the generalized inverse of a linear operator. Glas. Mat. 2 (1967), 265–271.Google Scholar