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2.4.1 Largest Eigenvalue and Related Eigenvector

Regard the $N$ eigenvectors $\vec{a}_{i}$ of $\mbox{${\bf A}$}$ as the base vectors in $N$-space. Then any $N$-vector may be written

\begin{displaymath}
\vec{x}_{0}= \sum_{i=1}^{N} c_{i} \, \vec{a}_{i}
\end{displaymath}

with appropriate coefficients $c_{i}$. Let $\vec{a}_{m}$ be the eigenvector corresponding to the largest (by absolute value) eigenvalue $\lambda_{m}$. Multiply $\vec{x}_{0}$ several times by $\mbox{${\bf A}$}$, each time normalizing the result:

\fbox{
\begin{minipage}{510pt}
\begin{displaymath}
\vec{x}_{k}\,'=\mbox{${\bf A}...
...k}\,'}{\left\vert \vec{x}_{k}\,' \right\vert }
\end{displaymath}\end{minipage}}

After a few iterations
\begin{displaymath}
\vec{x}_{k} \propto \sum_{i=1}^{N} c_{i} \, \lambda_{i}^{k} \, \vec{a}_{i}
\approx c_{m} \, \lambda_{m}^{k} \, \vec{a}_{m}
\end{displaymath} (2.5)

To obtain the eigenvalue $\lambda_{m}$, use

\fbox{
\begin{minipage}{510pt}
\begin{displaymath}
\lambda_{m} = \frac{{x_{\beta...
...
\lambda_{m} = \vec{x}_{k+1} \cdot \vec{x}_{k}'
\end{displaymath}\end{minipage}}

where ${x_{\beta}'}^{(k)}$ is any cartesian component of the unnormalized vector $\vec{x}_{k}\,'$.

EXAMPLE: Let

\begin{displaymath}
\mbox{${\bf A}$}=\mbox{$\left( \begin{array}{cc}3&1\\ \vspace{-9pt}\\ 2&4\end{array} \right)$}
\end{displaymath}

and use

\begin{displaymath}
\vec{x}_{0}=\mbox{$\left( \begin{array}{r} \sqrt{2}/2 \\ \vspace{-9 pt}\\ \sqrt{2}/2 \end{array} \right)$}
\end{displaymath}

choose as the starting vector. The iterated and normalized vectors are

\begin{displaymath}
\vec{x}_{1}=\mbox{$\left( \begin{array}{r} 0.555 \\ \vspace{...
....464 \\ \vspace{-9 pt}\\ 0.886 \end{array} \right)$}\; ; \dots
\end{displaymath}

From $\vec{x}_{3}$ and the unnormalized

\begin{displaymath}
\vec{x}_{4}\,'=\mbox{$\left( \begin{array}{r} 2.279 \\ \vspace{-9 pt}\\ 4.471 \end{array} \right)$}
\end{displaymath}

we find $\lambda_{m}=4.907$. The exact solution is

\begin{displaymath}
\vec{a}_{m}=\mbox{$\left( \begin{array}{r} 0.45 \\ \vspace{-...
...90 \end{array} \right)$} \;\; {\rm and} \;\; \lambda_{m}=5\, .
\end{displaymath}



next up previous
Next: 2.4.2 Arbitrary Eigenvalue/-vector: Inverse Up: 2.4 Eigenvalues and Eigenvectors Previous: 2.4 Eigenvalues and Eigenvectors
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001