Spectral radius and infinity norm of matrices
Let \(A=(a_{ij})\) be a real \(n\)-by-\(n\) matrix. Denote by \(\rho(A)\) the spectral radius and by \(\| A\| _{\infty}=\max_{1\leq i\leq n}\sum_{j=1}^{n}| a_{ij}| \) the infinity norm of \(A\). It is well known that \(\rho(A) \leq \| A\| _{\infty}\). (Actually the inequality holds for every matrix norm.) In the paper an algebraic criterion for \(\rho(A) < \| A\| _{\infty}\) is given. Namely, let \(a\) be a positive number and let \(A=(a_{ij})\) be a real \(n\)-by-\(n\) matrix. If \(\sum_{j=1}^{n}| a_{ij}| <a\) 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 \(\rho(A)<\| A\| _\infty\) if \(\varphi_{\| A\| _{\infty}}^{n}\bigl(A\bigr)=0\). 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.
scientific article; zbMATH DE number 5303208