Generalized eigenvectors and sets of nonnegative matrices. (English) Zbl 0548.15015

Zijm, W. H. M.

Linear Algebra Appl. 59, 91-113 (1984).

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

Keywords:

reducible; optimal choice property; dynamic programming; block triangular; generalized eigenvectors; square non-negative matrices; product property

Citations:

Zbl 0185.080; Zbl 0321.15010

References:

[1]	Burmeister, E.; Dobell, R., Mathematical Theories of Economic Growth (1970), MacMillan: MacMillan New York · Zbl 0238.90009
[2]	Dunford, N.; Schwartz, J. T., Linear Operators (1958), Interscience: Interscience New York, Part I · Zbl 0084.10402
[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: Springer Berlin, Heidelberg · Zbl 0117.13002
[5]	Howard, R. A., Dynamic Programming and Markov Processes (1960), Wiley: Wiley New York · Zbl 0091.16001
[6]	Karlin, S., A First Course in Stochastic Processes (1966), Academic: Academic New York · Zbl 0177.21102
[7]	Kemeny, J. G.; Snell, L. J., Finite Markov Chains (1960), Van Nostrand: 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: Allen and Unwin London · Zbl 0278.15011
[14]	Sladky, K., Successive approximation methods for dynamic programming models, Prague. Prague, Proceedings of the Third Formator Symposium on Mathematical Methods for the Analysis of Large Scale Systems, 171-189 (1979) · Zbl 0496.90081
[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.; 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, (Doctoral Thesis (1982), Eindhoven Univ. of Technology: Eindhoven Univ. of Technology Eindhoven) · Zbl 0526.90059
[19]	U.G. Rothblum, Growth decision problems, Abstract, in Official Program of the November 1975 ORSA/TIMS Meeting; U.G. Rothblum, Growth decision problems, Abstract, in Official Program of the November 1975 ORSA/TIMS Meeting
[20]	Rothblum, U. G.; Whittle, P., Growth optimality for branching Markov decision chains, Math. Oper. Res., 7, 582-601 (1982) · Zbl 0498.90082