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Totally expanding multiplicative systems. (English) Zbl 1078.15017
A single-matrix multiplicative system is given by the entries of the sequence \(\{Q^{n}x(0)\}_{n=0,1,\dots}\), where \(Q\) is an \(N \times N\) non-negative transition matrix and \(x(0)\) is an \(N \times 1\) semi-positive input vector. A single-matrix multiplicative system is said to be totally expanding if each coordinate of the sequence \((x(1), x(2), \dots)\) is unbounded.
Here, multiple-matrix multiplicative systems are studied. They are obtained when the single matrix \(Q\) is replaced by a set \(\{Q^{\delta}: \delta \in D\}\) of \(N \times N\) non-negative matrices, where \(D\) has a “product form” structure \(D= D_1 \times \dots \times D_N\), where for each \(i\), \(D_i\) is a finite non-empty set of non-negative \(1 \times N\) vectors. Each element \(\delta\) of \(D\) is called a policy. Such a system is said to be totally expanding if, for each policy \(\delta\), each coordinate of the sequence \((x^{\delta}(1), x^{\delta}(2), \dots)\) is unbounded.
It is shown that the multiple-matrix multiplicative system is totally expanding if and only if there are no “degenerate” coordinates, the growth rate \(\rho^{\delta}_{i} > 1\) for each \(\delta\) and each \(i\) and there exists an \(N \times 1\) vector \(u\) satisfying certain linear inequalities. These linear inequalities can give also the estimate of the smallest coordinate-dependent growth rate of the system.

MSC:
15B48 Positive matrices and their generalizations; cones of matrices
15A18 Eigenvalues, singular values, and eigenvectors
90B50 Management decision making, including multiple objectives
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