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matrix resolvent properties
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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$$ \|(sI-A)^{-1}\|\geq\frac{1}{\min\limits_{1\leq i\leq n}|s-\lambda_i|}=\frac{1}{\mathrm{dist}(s,\sigma_A)}.$$ 
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"matrix resolvent properties" is owned by Andrea Ambrosio.
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Cross-references: self consistent norm, eigenvalues, discrete set, point, complex, distance, self consistent matrix norm, property, simple, spectrum, proximity, norm, resolvent, matrix
This is version 12 of matrix resolvent properties, born on 2005-11-05, modified 2006-09-08.
Object id is 7468, canonical name is MatrixResolventProperties.
Accessed 2002 times total.
Classification:
| AMS MSC: | 15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions) |
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Pending Errata and Addenda
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