ResolventMatrix 2003-05-01 19:46:59 2006-12-30 16:16:20 Definition resolvent % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 $R_x(s)-R_y(s)=R_x(s)R_y(s)(x-y)$.