We consider a generic sequence of matrices (the nonnormal case is of interest) showing a proper cluster at zero in the sense of the singular values. By a direct use of the notion of majorizations, we show that the uniform spectral boundedness is sufficient for the proper clustering at zero of the eigenvalues: if the assumption of boundedness is removed, then we can construct sequences of matrices with a proper singular value clustering and having all the eigenvalues of an arbitrarily big modulus. Applications to the preconditioning theory are discussed.
Serra Capizzano, S., Bertaccini, D., Golub, G. (2005). How to Deduce a Proper Eigenvalue Cluster from a Proper Singular Value Cluster in the Nonnormal Case. SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 27(1), 82-86 [10.1137/040608027].
How to Deduce a Proper Eigenvalue Cluster from a Proper Singular Value Cluster in the Nonnormal Case
BERTACCINI, DANIELE;
2005-01-01
Abstract
We consider a generic sequence of matrices (the nonnormal case is of interest) showing a proper cluster at zero in the sense of the singular values. By a direct use of the notion of majorizations, we show that the uniform spectral boundedness is sufficient for the proper clustering at zero of the eigenvalues: if the assumption of boundedness is removed, then we can construct sequences of matrices with a proper singular value clustering and having all the eigenvalues of an arbitrarily big modulus. Applications to the preconditioning theory are discussed.File | Dimensione | Formato | |
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