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.
How to deduce a proper eigenvalue cluster from a proper singular value cluster in the nonnormal case
SERRA CAPIZZANO, STEFANO;
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.