# 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:

 $\|R_{A}(s)\|\geq\frac{1}{\mathrm{dist}(s,\sigma_{A})},$

where $\|.\|$ 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 $\mathrm{dist}(s,\sigma_{A})=\min\limits_{1\leq i\leq n}|s-\lambda_{i}|$.

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

 $\lim_{s\rightarrow\lambda_{i}}\|R_{A}(s)\|=+\infty.$

###### Proof.

Let ($\lambda_{i}$,$\mathbf{v}$) 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 $|\lambda|\leq\|A\|$, we have:

 $\max\limits_{1\leq i\leq n}\frac{1}{|s-\lambda_{i}|}\leq\|(sI-A)^{-1}\|$

whence

 $\|(sI-A)^{-1}\|\geq\frac{1}{\min\limits_{1\leq i\leq n}|s-\lambda_{i}|}=\frac{% 1}{\mathrm{dist}(s,\sigma_{A})}.$

Title matrix resolvent properties MatrixResolventProperties 2013-03-22 15:33:52 2013-03-22 15:33:52 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 15 Andrea Ambrosio (7332) Result msc 15A15