Maximum of the resolvent over matrices with given spectrum
Résumé
In numerical analysis it is often necessary to estimate the condition number $CN(T)=\left|\!\left|T\right|\!\right|_{} \cdot\left|\!\left|T^{-1}\right|\!\right|_{}$ and the norm of the resolvent $\left|\!\left|(\zeta-T)^{-1}\right|\!\right|_{}$ 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 $\left|\!\left|T\right|\!\right|_{} \leq1$ and minimal absolute eigenvalue $r=\min_{i=1,...,n}\left|\lambda_{i}\right|>0$ is the Kronecker bound $\frac{1}{r^{n}}$. This result is subsequently generalized by computing the corresponding supremum of $\left|\!\left|(\zeta-T)^{-1}\right|\!\right|_{}$ for any $\left|\zeta\right| \leq1$. 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.
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