Maximum of the resolvent over matrices with given spectrum

Abstract

In numerical analysis it is often necessary to estimate the condition number $CN(T)=||T|{|}_{}\cdot ||T^{-1}|{|}_{}$ and the norm of the resolvent $||(\zeta -T{)}^{-1}|{|}_{}$ of a given $n\times n$ matrix $T$. We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the well-known fact that the supremum of $CN(T)$ over all matrices with $||T|{|}_{}\le 1$ and minimal absolute eigenvalue $r={\min }_{i=1,...,n}|{\lambda }_i|>0$ is the Kronecker bound $\frac{1}{r^n}$. This result is subsequently generalized by computing the corresponding supremum of $||(\zeta -T{)}^{-1}|{|}_{}$ for any $|\zeta |\le 1$. We find that the supremum is attained by a triangular Toeplitz matrix. This provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically. The occuring Toeplitz matrices are so-called model matrices, i.e. matrix representations of the compressed backward shift operator on the Hardy space $H_2$ to a finite-dimensional invariant subspace.

References in corpus (1)

• Eigenvalue estimates for the resolvent of a non-normal matrix

Cited by in corpus (4)

• Spectral-Variation Bounds in Hyperbolic Geometry
• Infinite-dimensional features of matrices and pseudospectra
• $H^{\infty }$ interpolation and embedding theorems for rational functions
• Sharp estimate on the resolvent of a finite-dimensional contraction