Rachid Zarouf
I2M, Aix-Marseille Université
/user/rachid.zarouf/
Date(s) : 03/10/2016 iCal
11 h 00 min - 12 h 00 min
Maximum of the resolvent of an arbitrary matrix whose spectrum is fixed
In numerical analysis, it is often necessary to estimate the conditioning CN(T ) = 
and the norm of the resolvent 
of a given matrix T of size n×n. We give new spectral estimates for these quantities and explain matrices which allow us to reach our bounds. We find the following very old result (sometimes attributed to L. Kronecker): the upper bound of CN(T ) taken on all the matrices with a norm less than or equal to 1 and whose minimum of the eigenvalues (in absolute value) r = minλ
σ(T ){
} is strictly positive, is equal to 
. This result is then generalized by the calculation at ζ fixed in the closed unit disk, of the upper bound of 
, taken on all the matrices T of norm less than or equal to 1 whose spectrum σ(T ) of T is constrained to remain at a pseudo-hyperbolic distance at least r 
(0,1] from ζ. We note that this upper bound is reached by a triangular Toeplitz matrix. We thus provide a simple class of matrices whose conditioning or more generally the norm of The resolvent can be studied numerically. These extremal Toeplitz matrices are called model matrices since they are matrix representations of the compression of the left shift operator on the Hardy space H2 to an invariant subspace of finite dimension This work is carried out in collaboration with Oleg Szehr.
and the norm of the resolvent of a given matrix T of size n×n. We give new spectral estimates for these quantities and explain matrices which allow us to reach our bounds. We find the following very old result (sometimes attributed to L. Kronecker): the upper bound of CN(T ) taken on all the matrices with a norm less than or equal to 1 and whose minimum of the eigenvalues (in absolute value) r = minλσ(T ){} is strictly positive, is equal to . This result is then generalized by the calculation at ζ fixed in the closed unit disk, of the upper bound of , taken on all the matrices T of norm less than or equal to 1 whose spectrum σ(T ) of T is constrained to remain at a pseudo-hyperbolic distance at least r (0,1] from ζ. We note that this upper bound is reached by a triangular Toeplitz matrix. We thus provide a simple class of matrices whose conditioning or more generally the norm of The resolvent can be studied numerically. These extremal Toeplitz matrices are called model matrices since they are matrix representations of the compression of the left shift operator on the Hardy space H2 to an invariant subspace of finite dimension This work is carried out in collaboration with Oleg Szehr.https://hal.archives-ouvertes.fr/hal-01110346
Catégories