Nonnegative iterations with asymptotically constant coefficients

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2 Citations (Scopus)

Abstract

Let Ak, k = 0, 1, 2, ..., be a sequence of real nonsingular n × n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties A u = λ u and {norm of matrix} u {norm of matrix} = 1. Under further additional conditions on the spectrum of A, it is shown that if x0 ≠ 0 and the iteratesxk + 1 = Ak xk, k = 0, 1, 2, ...,are nonnegative, then frac(xk, {norm of matrix} xk {norm of matrix}) converges to u and frac({norm of matrix} xk + 1 {norm of matrix}, {norm of matrix} xk {norm of matrix}) converges to λ as k → ∞.

Original languageEnglish
Pages (from-to)1815-1824
Number of pages10
JournalLinear Algebra and Its Applications
Volume431
Issue number10
DOIs
Publication statusPublished - Oct 15 2009

Keywords

  • Convergence
  • Nonnegative iterations
  • z-Transform

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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