matrix resolvent properties

The matrix resolvent norm for a complex-valued $s$ is related to the proximity of such value to the spectrum of $A$; more precisely, the following simple yet meaningful property holds:

$$\parallel R_A(s)\parallel \ge \frac{1}{\mathrm{dist}(s,{\sigma }_A)},$$

where $\parallel .\parallel$ is any self consistent matrix norm, ${\sigma }_A$ is the spectrum of $A$ and the distance between a complex point and the discrete set of the eigenvalues ${\lambda }_i$ is defined as $\mathrm{dist}(s,{\sigma }_A)=\underset{1\le i\le n}{\min }|s-{\lambda }_i|$.

From this fact it comes immediately, for any $1\le i\le n$,

$$\underset{s\to {\lambda }_i}{lim}\parallel R_A(s)\parallel =+\mathrm{\infty }.$$

Proof.

Let (${\lambda }_i$,$𝐯$) be an eigenvalue-eigenvector pair of $A$; then

$$(sI-A)v=sv-Av=(s-{\lambda }_i)v$$

which shows $(s-{\lambda }_i)$ to be an eigenvalue of $(sI-A)$; ${(s-{\lambda }_i)}^{-1}$ is then an eigenvalue of ${(sI-A)}^{-1}$ and , since for any self consistent norm $|\lambda |\le \parallel A\parallel$, we have:

$$\underset{1\le i\le n}{\max }\frac{1}{|s-{\lambda }_i|}\le \parallel {(sI-A)}^{-1}\parallel$$

whence

$$\parallel {(sI-A)}^{-1}\parallel \ge \frac{1}{\underset{1\le i\le n}{\min }|s-{\lambda }_i|}=\frac{1}{\mathrm{dist}(s,{\sigma }_A)}.$$

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