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Spectral radius and infinity norm of matrices. (English) Zbl 1153.15016
Let $A=\left({a}_{ij}\right)$ be a real $n$-by-$n$ matrix. Denote by $\rho \left(A\right)$ the spectral radius and by ${\parallel A\parallel }_{\infty }={max}_{1\le i\le n}{\sum }_{j=1}^{n}|{a}_{ij}|$ the infinity norm of $A$. It is well known that $\rho \left(A\right)\le {\parallel A\parallel }_{\infty }$. (Actually the inequality holds for every matrix norm.) In the paper an algebraic criterion for ${\rho \left(A\right)<\parallel A\parallel }_{\infty }$ is given. Namely, let $a$ be a positive number and let $A=\left({a}_{ij}\right)$ be a real $n$-by-$n$ matrix. If ${\sum }_{j=1}^{n}|{a}_{ij}| for $i={i}_{1},...,{i}_{k}$, then substitute all elements in these $k$ rows and corresponding $k$ columns by zeros. Denote this transformation by ${\phi }_{a}$. One of the results in the paper says that ${\rho \left(A\right)<\parallel A\parallel }_{\infty }$ if ${\phi }_{{\parallel A\parallel }_{\infty }}^{n}\left(A\right)=0$. 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.
##### MSC:
 15A18 Eigenvalues, singular values, and eigenvectors 15A60 Applications of functional analysis to matrix theory 15A45 Miscellaneous inequalities involving matrices
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