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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 ρ(A) the spectral radius and by A =max 1in j=1 n |a ij | the infinity norm of A. It is well known that ρ(A)A . (Actually the inequality holds for every matrix norm.) In the paper an algebraic criterion for ρ(A)<A is given. Namely, let a be a positive number and let A=(a ij ) be a real n-by-n matrix. If j=1 n |a ij |<a for i=i 1 ,...,i k , then substitute all elements in these k rows and corresponding k columns by zeros. Denote this transformation by φ a . One of the results in the paper says that ρ(A)<A if φ A n A=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:
15A18Eigenvalues, singular values, and eigenvectors
15A60Applications of functional analysis to matrix theory
15A45Miscellaneous inequalities involving matrices
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