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The sign-real spectral radius and cycle products

โœ Scribed by Siegfried M. Rump

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
187 KB
Volume
279
Category
Article
ISSN
0024-3795

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โœฆ Synopsis

The extension of the Perron-Frobenius theory to real matrices without sign restriction uses the sign-real spectral radius as the generalization of the Perron root. The theory was used to extend and solve the conjecture in the affirmative that an ill-conditioned matrix is nearby a Singular matrix also in the componentwise sense. The proof estimates the ratio between the sign-real spectral radius and the maximum geometric mean of a cycle product. In this note we discuss bounds for this ratio including a counterexample to a conjecture about this ratio.

๐Ÿ“œ SIMILAR VOLUMES

Linear operators preserving the sign-rea
Linear operators preserving the sign-real spectral radius
โœ Bojana Zalar ๐Ÿ“‚ Article ๐Ÿ“… 1999 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 105 KB

Define the sign-real spectral radius of a real n ร— n matrix A as ฯ s 0 (A) = max SโˆˆS ฯ 0 (SA), where ฯ 0 (A) = max{|ฮป|; ฮป a real eigenvalue of A} is the real spectral radius of A and S denotes the set of signature matrices, i.e. S = {S; |S| = I}, the absolute value of matrices being meant entrywise.

On the spectral radius and the spectral
On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices
โœ Zejun Huang ๐Ÿ“‚ Article ๐Ÿ“… 2011 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 140 KB

## We prove the spectral radius inequality ฯ(A for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality A โ€ข B ฯ(A T B) for nonnegative matrices, which improves Schur's classical inequality , where โ€ข denotes the spectral norm. We also give counterexamples to two conject