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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Introduction to Eigenvectors\label{Introduction to Eigenvectors}}

\subsubsection{Eigenvectors\label{eigenvectors}}

Let $A$ be an $n\times n$ matrix. A nonzero vector $v$ is called an \textbf{%
eigenvector} of $A$ if there is a number $\lambda $ such that 
\[
Av=\lambda v 
\]
The number $\lambda $ is called an \textbf{eigenvalue\label{eigenvalue}}
associated with the eigenvector $v$. The subspace of $\UNICODE{0x211d} ^{n}$
consisting of all solutions to $Av=\lambda v$ is called the \textbf{%
eigenspace\label{eigenspace}} of $\lambda $ and is denoted by $E\left(
\lambda \right) $.

\subsubsection{Characteristic Equation\label{characteristic equation}}

Let $A$ be an $n\times n$ matrix. The function $p\left( \lambda \right)
=\det \left( A-\lambda I\right) $ is called the \textbf{characteristic
polynomial} of $A$ and the equation $\det \left( A-\lambda I\right) =0$ is
called the \textbf{characteristic equation} of $A$.

\begin{theorem}
$\lambda $ is an eigenvalue of a matrix $A$ if and only if $\det \left(
A-\lambda I\right) =0$. Each solution $\lambda $ ot the characteristic
equation has a corresponding eigenvector $x$: 
\[
\left( A-\lambda I\right) x=0\text{ or }Ax=\lambda x
\]
\end{theorem}

\begin{theorem}
(Cayley-Hamilton)\label{cayley-hamilton theorem} Any matrix satisfies its
own characteristic equation; that is, given a square matrix $A$ with real
eigenvalues, $\ p\left( A\right) =0$.
\end{theorem}

\subsubsection{Computing Eigenvectors\label{computing eigenvectors}}

Let $A$ be an $n\times n$ matrix. To find the eigenvectors of $A$

\begin{enumerate}
\item  Compute the determinant of $A-\lambda I$. This determinant is a
polynomial of degree $n$.

\item  Find the roots of this polynomial. The $n$ roots are the eigenvalues
of $A$.

\item  For each eigenvalue solve the equation $\left( A-\lambda I\right) x=0$%
. The solutions $x$ to this equation are the eigenvectors.\bigskip 
\end{enumerate}

\subsubsection{Trace and Determinant}

Let $A$ be an $n\times n$ matrix with eigenvalues $\lambda _{1},\lambda
_{2},...,\lambda _{n}$. Then the sum of the eigenvalues is the sum of the
diagonal entries of $A$: 
\[
\lambda _{1}+\lambda _{2}+...+\lambda _{n}=a_{11}+a_{22}+...+a_{nn}
\]
The sum of the diagonal entries of $A$ is called the \textbf{trace\label%
{trace}} of $A$, denoted $\limfunc{tr}A$. 

The product of the eigenvalues equals the determinant of $A$: 
\[
\lambda _{1}\lambda _{2}\cdots \lambda _{n}=\det A
\]

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Cayley-Hamilton theorem}{}{}{../../Help/DM8-6.tex#Minimum
polynomial}}

{\small \hyperref{Characteristic polynomial}{}{}{%
../../Help/DM8-6.tex#Characteristic, polynomial}}

{\small \hyperref{Determinant}{}{}{../../Help/DM8-5.tex#Determinant}}

{\small \hyperref{Eigenvalues and eigenvectors}{}{}{%
../../Help/DM8-6.tex#Eigenvectors}}

{\small \hyperref{Trace}{}{}{../../Help/DM8-5.tex#Trace}}

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