### Abstract

Let A_{k}, 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 x_{0} ≠ 0 and the iteratesx_{k + 1} = A_{k} x_{k}, k = 0, 1, 2, ...,are nonnegative, then frac(x_{k}, {norm of matrix} x_{k} {norm of matrix}) converges to u and frac({norm of matrix} x_{k + 1} {norm of matrix}, {norm of matrix} x_{k} {norm of matrix}) converges to λ as k → ∞.

Original language | English |
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Pages (from-to) | 1815-1824 |

Number of pages | 10 |

Journal | Linear Algebra and Its Applications |

Volume | 431 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 15 2009 |

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### Keywords

- Convergence
- Nonnegative iterations
- z-Transform

### ASJC Scopus subject areas

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

### Cite this

**Nonnegative iterations with asymptotically constant coefficients.** / Pituk, M.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 431, no. 10, pp. 1815-1824. https://doi.org/10.1016/j.laa.2009.06.020

}

TY - JOUR

T1 - Nonnegative iterations with asymptotically constant coefficients

AU - Pituk, M.

PY - 2009/10/15

Y1 - 2009/10/15

N2 - 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 → ∞.

AB - 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 → ∞.

KW - Convergence

KW - Nonnegative iterations

KW - z-Transform

UR - http://www.scopus.com/inward/record.url?scp=69349091594&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69349091594&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2009.06.020

DO - 10.1016/j.laa.2009.06.020

M3 - Article

VL - 431

SP - 1815

EP - 1824

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 10

ER -