# zbMATH — the first resource for mathematics

Generalized eigenvectors and sets of nonnegative matrices. (English) Zbl 0548.15015
Let K be the finite set of square non-negative matrices with index set $$\{$$ 1,2,...,$$N\}$$ with the ”product property”: that for each $$i=1,...,N$$ there exists a collection C(i) of non-negative row vectors of length N, and the elements of K are constructed by selecting the ith from C(i), $$i=1,...,N$$, all possible combinations being taken. The structural properties of the class K are studied in a manner developed from the approach of P. Mandl and the reviewer [Aust. J. Stat. 11, 85-96 (1969; Zbl 0185.080)] in a dynamic programming context; and the spectral approach of U. G. Rothblum [Linear Algebra Appl. 12, 281-292 (1975; Zbl 0321.15010)] for one matrix.
Reviewer: E.Seneta

##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 15A18 Eigenvalues, singular values, and eigenvectors 90C30 Nonlinear programming
Full Text:
##### References:
 [1] Burmeister, E.; Dobell, R., Mathematical theories of economic growth, (1970), MacMillan New York · Zbl 0238.90009 [2] Dunford, N.; Schwartz, J.T., Linear operators, (1958), Interscience New York, Part I [3] Gantmacher, F.R., The theory of matrices, Vol. II, (1959), (translated by K.A. Hirsch), Chelsea, New York · Zbl 0085.01001 [4] Harris, Th.E., The theory of branching processes, (1963), Springer Berlin, Heidelberg · Zbl 0117.13002 [5] Howard, R.A., Dynamic programming and Markov processes, (1960), Wiley New York · Zbl 0091.16001 [6] Karlin, S., A first course in stochastic processes, (1966), Academic New York · Zbl 0177.21102 [7] Kemeny, J.G.; Snell, L.J., Finite Markov chains, (1960), Van Nostrand Princeton, N.J · Zbl 0112.09802 [8] Mandl, P.; Seneta, E., The theory of non-negative matrices in a dynamic programming problem, Austral. J. statist., 11, 85-96, (1969) · Zbl 0185.08003 [9] Miller, B.L.; Veinott, A.F., Discrete dynamic programming with small interest rate, Ann. math. statist., 40, 366-370, (1966) · Zbl 0175.47302 [10] Rothblum, U.G., Algebraic eigenspaces of nonnegative matrices, Linear algebra appl., 12, 281-292, (1975) · Zbl 0321.15010 [11] Rothblum, U.G., Normalized Markov decision chains II: optimality of nonstationary policies, SIAM J. control. optim., 15, 2, 221-232, (1977) · Zbl 0367.90118 [12] Rothblum, U.G., Sensitive growth analysis of multiplicative systems I: the dynamic approach, report, (1979), Yale Univ [13] Seneta, E., Nonnegative matrices, (1973), Allen and Unwin London · Zbl 0278.15011 [14] Sladky, K., Successive approximation methods for dynamic programming models, Prague, Proceedings of the third formator symposium on mathematical methods for the analysis of large scale systems, 171-189, (1979) [15] Sladky, K., Bounds on discrete dynamic programming recursions I: models with nonnegative matrices, Kybernetica, 16, 526-547, (1980) · Zbl 0454.90085 [16] Sladky, K., Bounds on discrete dynamic programming recursions II: polynomial bounds on models with block-triangular structure, Kybernetica, 17, 310-328, (1981) · Zbl 0466.90084 [17] W.H.M. Zijm, Asymptotic expansions of dynamic programming recursions with general nonnegative matrices, submitted for publication. · Zbl 0595.90094 [18] Zijm, W.H.M., Nonnegative matrices in dynamic programming, () · Zbl 0526.90059 [19] U.G. Rothblum, Growth decision problems, Abstract, in Official Program of the November 1975 ORSA/TIMS Meeting, p. B-331. [20] Rothblum, U.G.; Whittle, P., Growth optimality for branching Markov decision chains, Math. oper. res., 7, 582-601, (1982) · Zbl 0498.90082
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.