Generalized $M$-matrices and applications
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- by George D. Poole PDF
- Math. Comp. 29 (1975), 903-910 Request permission
Abstract:
Recently, two distinct directions have been taken in an attempt to generalize the definition of an M-matrix. Even for nonsingular matrices, these two generalizations are not equivalent. The role of these and other classes of recently defined matrices is indicated showing their usefulness in various applications.References
- Abraham Berman and Robert J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math. 22 (1972), 155–161. MR 308139, DOI 10.1137/0122018
- Abraham Berman and Robert J. Plemmons, Cones and iterative methods for best least squares solutions of linear systems, SIAM J. Numer. Anal. 11 (1974), 145–154. MR 348984, DOI 10.1137/0711015
- Abraham Berman and Robert J. Plemmons, Matrix group monotonicity, Proc. Amer. Math. Soc. 46 (1974), 355–359. MR 352116, DOI 10.1090/S0002-9939-1974-0352116-0
- Thomas L. Boullion and Patrick L. Odell, Generalized inverse matrices, Wiley-Interscience [A Division of John Wiley & Sons, Inc.], New York-London-Sydney, 1971. MR 0338012
- David Carlson, A note on $M$-matrix equations, J. Soc. Indust. Appl. Math. 11 (1963), 1027–1033. MR 159830, DOI 10.1137/0111075
- R. E. Cline and R. J. Plemmons, $l_{2}$-solutions to underdetermined linear systems, SIAM Rev. 18 (1976), no. 1, 92–106. MR 396604, DOI 10.1137/1018004
- L. Collatz, Aufgaben monotoner Art, Arch. Math. 3 (1952), 366–376 (German). MR 53603, DOI 10.1007/BF01899376
- J. J. Dionísio, Non-negative vectors of a subspace of $R^{n}$ and positive solutions of linear systems, Univ. Lisboa Rev. Fac. Ci. A (2) 10 (1963/64), 165–177 (Portuguese). MR 179175 G. FROBENIUS, "Über Matrizen aus nicht negativen Elementen," S.-B. Preuss. Akad. Wiss. Berlin, v. 1912, pp. 456-477.
- O. L. Mangasarian, Characterizations of real matrices of monotone kind, SIAM Rev. 10 (1968), 439–441. MR 237537, DOI 10.1137/1010095
- O. L. Mangasarian, A convergent splitting of matrices, Numer. Math. 15 (1970), 351–353. MR 266409, DOI 10.1007/BF02165128
- O. L. Mangasarian, Perron-Frobenius properties of $Ax-\lambda Bx$, J. Math. Anal. Appl. 36 (1971), 86–102. MR 285555, DOI 10.1016/0022-247X(71)90020-5
- Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Inc., Boston, Mass., 1964. MR 0162808
- L. Negrescu, On systems of linear inequalities with non-negative solutions. Applications to linear programming, Com. Acad. R. P. Romîne 13 (1963), 761–764 (Romanian, with French and Russian summaries). MR 181467
- Liviu Negrescu, On some systems of inequalities and linear equations with non-negative solutions. Applications to linear programming, Acad. R. P. Romîne Fil. Cluj Stud. Cerc. Mat. 14 (1963), 93–102 (Romanian, with French and Russian summaries). MR 190154
- Alexander Ostrowski, Über die determinanten mit überwiegender Hauptdiagonale, Comment. Math. Helv. 10 (1937), no. 1, 69–96 (German). MR 1509568, DOI 10.1007/BF01214284
- R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406–413. MR 69793, DOI 10.1017/S0305004100030401
- Robert J. Plemmons, Monotonicity and iterative approximations involving rectangular matrices, Math. Comp. 26 (1972), 853–858. MR 315882, DOI 10.1090/S0025-5718-1972-0315882-2 ROBERT J. PLEMMONS, "Convergent splittings for best approximate solutions to linear systems."
- R. J. Plemmons, Direct iterative methods for linear systems using weak splittings, Acta Univ. Carolin. Math. Phys. 15 (1974), no. 1-2, 117–120. MR 388751
- Robert J. Plemmons, Linear least squares by elimination and MGS, J. Assoc. Comput. Mach. 21 (1974), 581–585. MR 356474, DOI 10.1145/321850.321855
- Robert J. Plemmons, Linear least squares by elimination and MGS, J. Assoc. Comput. Mach. 21 (1974), 581–585. MR 356474, DOI 10.1145/321850.321855
- George Poole and Thomas Boullion, A survey on $M$-matrices, SIAM Rev. 16 (1974), 419–427. MR 352146, DOI 10.1137/1016079
- Hans Schneider, The elementary divisors, associated with $0$, of a singular $M$-matrix, Proc. Edinburgh Math. Soc. (2) 10 (1956), 108–122. MR 76725, DOI 10.1017/S0013091500021507
- T. J. Stieltjes, Sur les racines de l’équation $X^n=0$, Acta Math. 9 (1887), no. 1, 385–400 (French). MR 1554723, DOI 10.1007/BF02406744
- W. Tutschke, Eine hinreichende Bedingung für die Existenz positiver Lösungen von linearen Gleichungssystemen, Monatsb. Deutsch. Akad. Wiss. Berlin 5 (1963), 663–667 (German). MR 166210
- Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Math. Comp. 29 (1975), 903-910
- MSC: Primary 15A09
- DOI: https://doi.org/10.1090/S0025-5718-1975-0369384-0
- MathSciNet review: 0369384