MultiplicityOfEigenvalue 2005-05-10 15:02:11 2007-07-08 12:30:56 Definition eigenvalue geometric multiplicity algebraic multiplicity eigenvalue % 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} \usepackage{amsthm} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions % % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\F}{\mathbbmss{F}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} Suppose $V$ is a finite dimensional vector space over a field $\F$, and suppose $L\colon V\to V$ is a linear map. Suppose also that $\lambda\in \F$ is an eigenvalue of $L$, that is, $\operatorname{det}( L - \lambda I)=0$. The \emph{algebrai{c} multiplicit{y}}, denoted by $A_\lambda(L)$, of $\lambda$ is the multiplicity of the root $\lambda$ to the polynomial $\operatorname{det}( L - \lambda I)=0$. The \emph{geometric multiplicity} of $\lambda$, denoted by $G_\lambda(L)$, is the dimension of $\ker ( L - \lambda I)$, the eigenspace of $\lambda$.