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Spectral radius and infinity norm of matrices. (English) Zbl 1153.15016
Let $A=(a_{ij})$ be a real $n$-by-$n$ matrix. Denote by $\rho(A)$ the spectral radius and by $\Vert A\Vert _{\infty}=\max_{1\leq i\leq n}\sum_{j=1}^{n}\vert a_{ij}\vert $ the infinity norm of $A$. It is well known that $\rho(A) \leq \Vert A\Vert _{\infty}$. (Actually the inequality holds for every matrix norm.) In the paper an algebraic criterion for $\rho(A) < \Vert A\Vert _{\infty}$ is given. Namely, let $a$ be a positive number and let $A=(a_{ij})$ be a real $n$-by-$n$ matrix. If $\sum_{j=1}^{n}\vert a_{ij}\vert <a$ for $i=i_1,\ldots, i_k$, then substitute all elements in these $k$ rows and corresponding $k$ columns by zeros. Denote this transformation by $\varphi_a$. One of the results in the paper says that $\rho(A)<\Vert A\Vert _\infty$ if $\varphi_{\Vert A\Vert _{\infty}}^{n}\bigl(A\bigr)=0$. On the other hand, if $A$ is entrywise nonnegative, the converse holds as well. There are some other related results and an application to the discrete dynamical systems.

15A18Eigenvalues, singular values, and eigenvectors
15A60Applications of functional analysis to matrix theory
15A45Miscellaneous inequalities involving matrices
Full Text: DOI
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