Maximum of the resolvent over matrices with given spectrum

TL;DR

This paper derives new spectral bounds for the condition number and resolvent norm of matrices with given spectra, identifying explicit extremal matrices and generalizing classical bounds in numerical analysis.

Contribution

It introduces new spectral estimates for the condition number and resolvent norm, explicitly characterizes extremal matrices, and generalizes known bounds for matrices with specified spectra.

Findings

Supremum of the condition number over matrices with norm ≤ 1 and minimal eigenvalue r is 1/r^n.

Supremum of the resolvent norm for |ζ| ≤ 1 is attained by a triangular Toeplitz matrix.

Explicit extremal matrices are identified as structured model matrices related to the Hardy space.

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 $||T||_{} \leq1$ 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 $|\zeta| \leq1$. 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.