# resolvent matrix

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

If $A$ is commutative and $s\notin Sp(x)\cup Sp(y)$, then $R_{x}(s)-R_{y}(s)=R_{x}(s)R_{y}(s)(x-y)$.

Title resolvent matrix ResolventMatrix 2013-03-22 13:36:20 2013-03-22 13:36:20 mps (409) mps (409) 8 mps (409) Definition msc 47A10 msc 15A15 resolvent