×

Generalized inverses of matrices: a perspective of the work of Penrose. (English) Zbl 0616.15003

E. H. Moore’s ”general reciprocal” (announced about 50 years ahead of Penrose) remained unknown mainly because of a special hard-to-read notation. Here it is shown that Moore’s object is equivalent to the more familiar definitions of (what later became) the Moore-Penrose generalized inverse. A concise, but easy-to-read, survey of the main ideas and applications of the generalized inverse is given. This is an important paper for the historians of generalized inverses and for the instructors in linear algebra.
Reviewer: S.Zlobec

MSC:

15A09 Theory of matrix inversion and generalized inverses
15-03 History of linear algebra
15A06 Linear equations (linear algebraic aspects)
01A60 History of mathematics in the 20th century
01A65 Development of contemporary mathematics
15A18 Eigenvalues, singular values, and eigenvectors
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Strang, Linear Algebra and its Applications (1976)
[2] DOI: 10.1137/1019104 · Zbl 0379.65021 · doi:10.1137/1019104
[3] DOI: 10.1137/1013063 · Zbl 0199.20503 · doi:10.1137/1013063
[4] DOI: 10.2307/2308576 · Zbl 0083.02901 · doi:10.2307/2308576
[5] Searle, Matrix Algebra useful for Statistics (1982) · Zbl 0555.62002
[6] DOI: 10.1016/0024-3795(71)90002-4 · Zbl 0254.15004 · doi:10.1016/0024-3795(71)90002-4
[7] DOI: 10.2307/2312140 · Zbl 0106.01602 · doi:10.2307/2312140
[8] Davis, Linear Algebra Appl 5 pp 329– (1972)
[9] DOI: 10.1016/0022-247X(78)90028-8 · Zbl 0379.93017 · doi:10.1016/0022-247X(78)90028-8
[10] Cline, SIAM J. Appl. Math 5 pp 329– (1972)
[11] DOI: 10.1016/0024-3795(80)90230-X · Zbl 0433.15002 · doi:10.1016/0024-3795(80)90230-X
[12] DOI: 10.1016/0024-3795(81)90137-3 · Zbl 0468.15003 · doi:10.1016/0024-3795(81)90137-3
[13] DOI: 10.1137/0705015 · Zbl 0165.34602 · doi:10.1137/0705015
[14] DOI: 10.1016/0022-247X(68)90204-7 · Zbl 0159.32101 · doi:10.1016/0022-247X(68)90204-7
[15] DOI: 10.1137/0124050 · Zbl 0237.15001 · doi:10.1137/0124050
[16] Rao, Generalized Inverse of Matrices and its Applications (1971) · Zbl 0236.15004
[17] Rao, Sankhy? Ser. A 29 pp 317– (1967)
[18] Rao, J. Roy. Statist. Soc 24 pp 152– (1962)
[19] DOI: 10.2307/2282625 · Zbl 0144.42401 · doi:10.2307/2282625
[20] Rado, Proc. Cambridge Philos. Soc 52 pp 600– (1956)
[21] DOI: 10.1214/aoms/1177728915 · Zbl 0053.10504 · doi:10.1214/aoms/1177728915
[22] Charnes, Naval Res. Logist. Quart 15 pp 517– (1968) · Zbl 0177.48102 · doi:10.1002/nav.3800150405
[23] DOI: 10.1137/0122032 · Zbl 0244.90026 · doi:10.1137/0122032
[24] DOI: 10.1137/0131035 · Zbl 0341.34001 · doi:10.1137/0131035
[25] Campbell, Generalized Inverses of Linear Transformations (1979)
[26] Campbell, Recent Applications of Generalized Inverses (1982)
[27] DOI: 10.1080/00207727808941742 · Zbl 0384.49013 · doi:10.1080/00207727808941742
[28] DOI: 10.1137/0508081 · Zbl 0379.34009 · doi:10.1137/0508081
[29] DOI: 10.1016/0024-3795(81)90145-2 · Zbl 0477.15003 · doi:10.1016/0024-3795(81)90145-2
[30] Pringle, Generalized Inverse Matrices with Applications to Statistics (1971)
[31] DOI: 10.1016/0022-247X(66)90048-5 · Zbl 0151.13002 · doi:10.1016/0022-247X(66)90048-5
[32] DOI: 10.1016/0022-247X(66)90088-6 · Zbl 0161.29102 · doi:10.1016/0022-247X(66)90088-6
[33] DOI: 10.1007/BF01691464 · Zbl 0215.21803 · doi:10.1007/BF01691464
[34] Plemmons, Proc. Cambridge Philos. Soc 31 pp 46– (1972)
[35] Englefield, Proc. Cambridge Philos. Soc 62 pp 667– (1966)
[36] DOI: 10.1016/0022-247X(81)90217-1 · Zbl 0492.47012 · doi:10.1016/0022-247X(81)90217-1
[37] DOI: 10.1090/S0002-9904-1939-06910-3 · Zbl 0020.19802 · doi:10.1090/S0002-9904-1939-06910-3
[38] DOI: 10.1007/BF02288367 · JFM 62.1075.02 · doi:10.1007/BF02288367
[39] Drygas, The Coordinate-Free Approach to Gauss-Markov Estimation (1970) · Zbl 0215.26504 · doi:10.1007/978-3-642-65148-9
[40] DOI: 10.1016/0024-3795(81)90101-4 · Zbl 0471.15007 · doi:10.1016/0024-3795(81)90101-4
[41] DOI: 10.1016/0024-3795(81)90149-X · Zbl 0466.15009 · doi:10.1016/0024-3795(81)90149-X
[42] DOI: 10.1080/03081087308817015 · Zbl 0291.15004 · doi:10.1080/03081087308817015
[43] Hartwig, Pacific J. Math 69 pp 73– (1977) · Zbl 0326.16012 · doi:10.2140/pjm.1977.69.73
[44] DOI: 10.1016/0024-3795(81)90181-6 · Zbl 0455.15003 · doi:10.1016/0024-3795(81)90181-6
[45] Groetsch, Generalized Inverses of Linear Operators. Representation and Approximation (1977) · Zbl 0358.47001
[46] DOI: 10.1016/0024-3795(73)90021-9 · Zbl 0247.15004 · doi:10.1016/0024-3795(73)90021-9
[47] Munn, Proc. Cambridge Philos. Soc 51 pp 396– (1955)
[48] Munn, Proc. Cambridge Philos. Soc 57 pp 247– (1961)
[49] DOI: 10.1137/0137016 · Zbl 0438.62055 · doi:10.1137/0137016
[50] Moore, General Analysis, Part II. The Fundamental Notions of General Analysis (1939) · JFM 65.0497.05
[51] Moore, General Analysis, Part I. The Algebra of Matrices (1935)
[52] Moore, Bull. Amer. Math. Soc 26 pp 394– (1920)
[53] DOI: 10.1103/PhysRev.80.81 · Zbl 0041.57305 · doi:10.1103/PhysRev.80.81
[54] Minamide, J. Math. Anal. Appl 35 pp 222– (1971)
[55] DOI: 10.1137/0119015 · Zbl 0299.15003 · doi:10.1137/0119015
[56] Ben-Israel, Generalized Inverses: Theory and Applications (1974)
[57] DOI: 10.1137/0116075 · Zbl 0167.30304 · doi:10.1137/0116075
[58] Ben-Israel, Oper. Res 16 pp 1167– (1968)
[59] DOI: 10.1016/0022-247X(67)90067-4 · doi:10.1016/0022-247X(67)90067-4
[60] DOI: 10.1016/0022-247X(63)90064-7 · Zbl 0203.15106 · doi:10.1016/0022-247X(63)90064-7
[61] Zlobec, Glas. Mat 2 pp 265– (1967)
[62] DOI: 10.1137/0111051 · Zbl 0116.32202 · doi:10.1137/0111051
[63] DOI: 10.1007/BF02163027 · Zbl 0181.17602 · doi:10.1007/BF02163027
[64] DOI: 10.1016/0001-8708(68)90021-2 · Zbl 0189.14805 · doi:10.1016/0001-8708(68)90021-2
[65] DOI: 10.1016/0024-3795(82)90255-5 · Zbl 0487.15004 · doi:10.1016/0024-3795(82)90255-5
[66] Golub, SIAM J. Numer. Anal 2 pp 205– (1965)
[67] Golub, Apl. Mat 13 pp 44– (1968)
[68] Goldman, J. Res. Nat. Bur. Standards Sect. B 68B pp 151– (1964) · Zbl 0127.35703 · doi:10.6028/jres.068B.021
[69] DOI: 10.1016/0024-3795(86)90210-7 · Zbl 0583.60064 · doi:10.1016/0024-3795(86)90210-7
[70] DOI: 10.1007/BF02421317 · JFM 34.0422.02 · doi:10.1007/BF02421317
[71] DOI: 10.1307/mmj/1028998825 · Zbl 0116.25404 · doi:10.1307/mmj/1028998825
[72] DOI: 10.1093/comjnl/10.4.392 · Zbl 0155.19803 · doi:10.1093/comjnl/10.4.392
[73] DOI: 10.1137/1017044 · Zbl 0313.60044 · doi:10.1137/1017044
[74] DOI: 10.1090/S0002-9904-1939-06933-4 · Zbl 0020.20001 · doi:10.1090/S0002-9904-1939-06933-4
[75] MacDuffee, The Theory of Matrices (1956)
[76] DOI: 10.1016/0016-0032(75)90002-2 · Zbl 0321.34007 · doi:10.1016/0016-0032(75)90002-2
[77] DOI: 10.1016/0024-3795(82)90114-8 · Zbl 0505.15003 · doi:10.1016/0024-3795(82)90114-8
[78] DOI: 10.1080/00207177608932859 · Zbl 0336.93011 · doi:10.1080/00207177608932859
[79] DOI: 10.2307/2006345 · Zbl 0406.65028 · doi:10.2307/2006345
[80] DOI: 10.1080/00207177608932835 · doi:10.1080/00207177608932835
[81] DOI: 10.1137/0511052 · Zbl 0448.47023 · doi:10.1137/0511052
[82] DOI: 10.1016/0022-247X(66)90115-6 · Zbl 0139.10301 · doi:10.1016/0022-247X(66)90115-6
[83] DOI: 10.1007/BF01933494 · Zbl 0263.65047 · doi:10.1007/BF01933494
[84] DOI: 10.1137/1012042 · Zbl 0196.16001 · doi:10.1137/1012042
[85] Autonne, Ann. Univ. Lyon Sect. A 38 pp 1– (1915)
[86] Wedderburn, Lectures on Matrices XVII (1934) · doi:10.1090/coll/017
[87] DOI: 10.1137/0114030 · Zbl 0142.00303 · doi:10.1137/0114030
[88] Atkinson, Acta Sci. Math 15 pp 38– (1953)
[89] DOI: 10.1016/0024-3795(82)90117-3 · Zbl 0501.15004 · doi:10.1016/0024-3795(82)90117-3
[90] Anselone, Approximation Theory III pp 163– (1980)
[91] Tseng, Uspehi Mat. Nauk (N.S.) 11 pp 213– (1956)
[92] DOI: 10.1016/0022-247X(69)90200-5 · Zbl 0177.04904 · doi:10.1016/0022-247X(69)90200-5
[93] Fiacco, Extremal Methods and Systems Analysis 174 (1980) · doi:10.1007/978-3-642-46414-0_2
[94] Tseng, Dokl. Akad. Nauk SSSR (N.S.) 67 pp 607– (1949)
[95] DOI: 10.1016/0022-247X(72)90202-8 · Zbl 0266.47012 · doi:10.1016/0022-247X(72)90202-8
[96] Tseng, Dokl. Akad. Nauk SSSR (N.S.) 67 pp 431– (1949)
[97] Albert, Regression and the Moore-Penrose Pseudoinverse (1972)
[98] Erdelyi, Proc. Cambridge Philos. Soc 71 pp 43– (1972)
[99] Tseng, C. R. Acad. Sci 228 pp 640– (1949)
[100] Afriat, Proc. Cambridge Philos. Soc 53 pp 800– (1957)
[101] DOI: 10.1137/0314068 · Zbl 0346.49010 · doi:10.1137/0314068
[102] DOI: 10.1093/comjnl/13.3.309 · Zbl 0195.44804 · doi:10.1093/comjnl/13.3.309
[103] DOI: 10.1137/0130062 · Zbl 0329.65027 · doi:10.1137/0130062
[104] Penrose, Proc. Cambridge Philos. Soc 52 pp 17– (1956)
[105] Cajori, A History of Mathematical Notation, Volume II. Notations mainly in Higher Mathematics (1929) · JFM 55.0002.02
[106] Penrose, Proc. Cambridge Philos. Soc 51 pp 406– (1955)
[107] DOI: 10.1137/0126008 · Zbl 0243.15004 · doi:10.1137/0126008
[108] Boullion, Generalized Inverse Matrices (1971)
[109] DOI: 10.1109/TAC.1981.1102599 · Zbl 0475.93029 · doi:10.1109/TAC.1981.1102599
[110] Boullion, Proceedings of the Symposium on Theory and Applications of Generalized Inverses of Matrices (1968)
[111] Laurent, Approximation et Optimisation (1972) · Zbl 0238.90058
[112] DOI: 10.2307/1990850 · Zbl 0050.25104 · doi:10.2307/1990850
[113] DOI: 10.1137/0115105 · Zbl 0155.35406 · doi:10.1137/0115105
[114] DOI: 10.2307/2005322 · Zbl 0342.65025 · doi:10.2307/2005322
[115] Landesman, Pacific J. Math 21 pp 113– (1967) · Zbl 0146.34303 · doi:10.2140/pjm.1967.21.113
[116] DOI: 10.1090/S0002-9904-1934-05872-5 · doi:10.1090/S0002-9904-1934-05872-5
[117] DOI: 10.1016/0022-247X(77)90194-9 · Zbl 0357.65034 · doi:10.1016/0022-247X(77)90194-9
[118] DOI: 10.1090/S0002-9904-1933-05727-0 · doi:10.1090/S0002-9904-1933-05727-0
[119] Kishi, Advances in Control Systems Theory and Applications pp 245– (1964)
[120] Kalman, Trans. ASME Ser. D: J. Basic Engrg 83 pp 95– (1961) · doi:10.1115/1.3658902
[121] Kalman, Trans. ASME Ser. D: J. Basic Engrg 82 pp 35– (1960) · doi:10.1115/1.3662552
[122] DOI: 10.2307/1988578 · JFM 43.0425.04 · doi:10.2307/1988578
[123] DOI: 10.1016/0024-3795(68)90028-1 · Zbl 0186.33602 · doi:10.1016/0024-3795(68)90028-1
[124] Noble, Applied Linear Algebra (1977)
[125] DOI: 10.1137/0117050 · Zbl 0191.42202 · doi:10.1137/0117050
[126] DOI: 10.1073/pnas.22.12.707 · Zbl 0015.38802 · doi:10.1073/pnas.22.12.707
[127] DOI: 10.2307/2005662 · Zbl 0282.65031 · doi:10.2307/2005662
[128] Bjerhammar, A Theory of Errors and Generalized Inverse Matrices (1973) · Zbl 0267.65002
[129] Bjerhammar, Svensk Lantm?teritidskrift h?fte 5?6 pp 311– (1955)
[130] Nashed, Generalized Inverses and Applications (1976)
[131] DOI: 10.1007/BF02526278 · doi:10.1007/BF02526278
[132] DOI: 10.1137/0119053 · Zbl 0236.15009 · doi:10.1137/0119053
[133] DOI: 10.1016/0022-247X(65)90109-5 · doi:10.1016/0022-247X(65)90109-5
[134] Hoerl, Chem. Engng. Prog 58 pp 54– (1962)
[135] DOI: 10.1016/0022-247X(65)90108-3 · Zbl 0151.19205 · doi:10.1016/0022-247X(65)90108-3
[136] Hilbert, Grundz?ge einer allgemeinen Theorie der linearen Integralgleichungen pp 49– (1912) · JFM 43.0423.01
[137] DOI: 10.1007/BF00411592 · Zbl 0178.00503 · doi:10.1007/BF00411592
[138] DOI: 10.1007/BF00412288 · Zbl 0149.33703 · doi:10.1007/BF00412288
[139] DOI: 10.1137/0122018 · Zbl 0255.15005 · doi:10.1137/0122018
[140] DOI: 10.1007/BF00253936 · Zbl 0201.37001 · doi:10.1007/BF00253936
[141] DOI: 10.1137/0120046 · Zbl 0227.05013 · doi:10.1137/0120046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.