Let A be a nonnegative $n n$ matrix. In this paper we study the growth of the powers $A^m, m = 1,2,3, $ when $( A ) = 1$. These powers occur naturally in the iteration process \x^( m + 1 ) = Ax^( m ) ,x^( 0 ) 0,\ which is important in applications and numerical techniques. Roughly speaking, we analyze the asymptotic behavior of each entry of $A^m $. We apply our main result to determine necessary and sufficient conditions for the convergence to the spectral radius of A of certain ratios naturally associated with the iteration above.
%0 Journal Article
%1 friedland1980growth
%A Friedland, Shmuel
%A Schneider, Hans
%D 1980
%J SIAM Journal on Algebraic Discrete Methods
%K eigenvalues matrix nonnegative
%N 2
%P 185--200
%R 10.1137/0601022
%T The Growth of Powers of a Nonnegative Matrix
%V 1
%X Let A be a nonnegative $n n$ matrix. In this paper we study the growth of the powers $A^m, m = 1,2,3, $ when $( A ) = 1$. These powers occur naturally in the iteration process \x^( m + 1 ) = Ax^( m ) ,x^( 0 ) 0,\ which is important in applications and numerical techniques. Roughly speaking, we analyze the asymptotic behavior of each entry of $A^m $. We apply our main result to determine necessary and sufficient conditions for the convergence to the spectral radius of A of certain ratios naturally associated with the iteration above.
@article{friedland1980growth,
abstract = {Let A be a nonnegative $n \times n$ matrix. In this paper we study the growth of the powers $A^m, m = 1,2,3, \cdots $ when $\rho ( A ) = 1$. These powers occur naturally in the iteration process \[x^{( m + 1 )} = Ax^{( m )} ,\quad x^{( 0 )} \geqq 0,\] which is important in applications and numerical techniques. Roughly speaking, we analyze the asymptotic behavior of each entry of $A^m $. We apply our main result to determine necessary and sufficient conditions for the convergence to the spectral radius of A of certain ratios naturally associated with the iteration above.},
added-at = {2017-02-01T08:12:23.000+0100},
author = {Friedland, Shmuel and Schneider, Hans},
biburl = {https://www.bibsonomy.org/bibtex/2106121ee3e37ac5372ac1127b9cd1ab2/ytyoun},
doi = {10.1137/0601022},
interhash = {7ddde2ceac3805db945b280c25817c05},
intrahash = {106121ee3e37ac5372ac1127b9cd1ab2},
journal = {{SIAM} Journal on Algebraic Discrete Methods},
keywords = {eigenvalues matrix nonnegative},
number = 2,
pages = {185--200},
timestamp = {2017-11-21T05:02:54.000+0100},
title = {The Growth of Powers of a Nonnegative Matrix},
volume = 1,
year = 1980
}