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Generalized inverses of substochastic matrices. (English) Zbl 0619.15004

Author’s summary: This paper looks at the question of when a substochastic matrix has a substochastic generalized inverse. This question is answered for several generalized inverses, including semiinverses, the Moore-Penrose inverse, and the group inverse. Methods for constructing all such inverses are given.
Reviewer: S.L.Campbell

MSC:

15A09 Theory of matrix inversion and generalized inverses
15B51 Stochastic matrices
Full Text: DOI

References:

[1] Ben-Israel, A.; Charnes, A., Contributions to the theory of generalized inverses, J. Soc. Indust. Appl. Math., XI, 667-699 (1963) · Zbl 0116.32202
[2] Ben-Israel, A.; Greville, T. N.E., Generalized Inverses: Theory and Applications (1974), Wiley-Interscience: Wiley-Interscience New York · Zbl 0305.15001
[3] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic: Academic New York · Zbl 0484.15016
[4] Doob, J. L., Topics in the theory of Markov chains, Trans. Amer. Math. Soc., 52, 37-64 (1942) · Zbl 0063.09001
[5] Farahat, H. K., The semigroup of doubly-stochastic matrices, Proc. Glasgow Math. Assoc., 7, 178-183 (1966) · Zbl 0156.26001
[6] Greville, T. N.E., Some applications of the pseudoinverse of a matrix, SIAM Rev., II, 15-22 (1960) · Zbl 0168.13303
[7] Haynsworth, E. V.; Wall, J. R., Group inverses of certain nonnegative matrices, Linear Algebra Appl., 25, 271-288 (1979) · Zbl 0403.15005
[8] Penrose, R., A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51, 406-413 (1955) · Zbl 0065.24603
[9] Plemmons, R. J.; Cline, R. E., The generalized inverse of a nonnegative matrix, Proc. Amer. Math. Soc., 31, 46-50 (1972) · Zbl 0241.15001
[10] Robinson, C. E., Green’s relations for substochastic matrices, Linear Algebra Appl., 80, 39-53 (1986) · Zbl 0599.15012
[11] Schwarz, S., On the structure of the semigroup of stochastic matrices, Magyar Tud. Akad. Kutato Int. Kozl. (A), 9, 297-311 (1964) · Zbl 0143.03303
[12] Schwarz, S., A note on the structure of the semigroup of doubly stochastic matrices, Mat. C̆asopis Sloven. Akad. Vied., 17, 308-316 (1967) · Zbl 0157.04902
[13] Tam, B. S., A geometric treatment of generalized inverses and semigroups of nonnegative matrices, Linear Algebra Appl., 41, 225-272 (1981) · Zbl 0471.15007
[14] Wall, J. R., Generalized inverses of stochastic matrices, Linear Algebra Appl., 10, 147-154 (1975) · Zbl 0305.15002
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