PlanetMath: resolvent matrix

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resolvent matrix

(Definition)

The resolvent matrix of a matrix is defined as

$\displaystyle R_{A}(s)=(sI-A)^{-1}.$

Note: is the identity matrix and is a complex variable. Also note that $R_{A}(s)$ is undefined on (the spectrum of ).

More generally, let be a unital algebra over the field of complex numbers $\mathbb{C}$ . The resolvent of an element $x\in A$ is a function from $\mathbb{C}-Sp(x)$ to given by

$\displaystyle R_x(s)=(s\cdot 1-x)^{-1}$

where

is the spectrum of

: $Sp(x)=\lbrace t\in \mathbb{C}\mid t\cdot 1 -x$ is not invertible in $A\rbrace$ .

If is commutative and $s\notin Sp(x)\cup Sp(y)$ , then .

"resolvent matrix" is owned by mps. [ full author list (3) | owner history (1) ]

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Also defines:

resolvent

Attachments:: matrix resolvent properties (Result) by Andrea Ambrosio
resolvent function is analytic (Theorem) by asteroid

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Cross-references: commutative, function, complex numbers, field, algebra, unital, spectrum, variable, complex, identity matrix, matrix
There are 7 references to this entry.

This is version 5 of resolvent matrix, born on 2003-05-01, modified 2006-12-30.
Object id is 4235, canonical name is ResolventMatrix.
Accessed 4201 times total.

Classification:

AMS MSC:	15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)
	47A10 (Operator theory :: General theory of linear operators :: Spectrum, resolvent)

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