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'resolvent matrix'
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| Title of object: |
resolvent matrix |
| Canonical Name: |
ResolventMatrix |
| Type: |
Definition |
| Created on: |
2003-05-01 19:46:59 |
| Modified on: |
2006-12-30 13:54:48 |
| Classification: |
msc:15A15, msc:47A10 |
| Defines: |
resolvent |
Preamble:
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Content:
The \emph{resolvent matrix} of a matrix $A$ is defined as
\[
R_{A}(s)=(sI-A)^{-1}.
\]
Note: $I$ is the identity matrix and $s$ is a complex variable. Also note that $R_{A}(s)$ is undefined on $Sp(A)$ (the spectrum of $A$).
More generally, let $A$ be a unital algebra over the field of complex numbers $\mathbb{C}$. The \emph{resolvent} $R_x$ of an element $x\in A$ is a function from $\mathbb{C}-Sp(x)$ to $A$ given by
\[
R_x(s)=(s\cdot 1-x)^{-1}
\]
where $Sp(x)$ is the spectrum of $x$: $Sp(x)=\lbrace t\in \mathbb{C}\mid t\cdot 1 -x\mbox{ is not invertible in }A\rbrace$.
If $A$ is commutative and $s\notin Sp(x)\cup Sp(y)$, then $R_x(s)R_y(s)=R_x(s)R_y(s)(x-y)$. |
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