%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 }