Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag

Mathematics-Online lexicon:

Deflation


A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

If $ \lambda$ is an eigenvalue of an $ n \times n$ matrix $ A$ with an eigenvector of the form $ \left( 1,\, u_1,\, \hdots ,\, u_{n-1} \right)$, then

$\displaystyle \underbrace{\left( \begin{array}{c\vert ccc}
1 & 0 & \cdots & 0 \...
...A(1,2:n) & \\ \hline
0 & & & \\
\vdots & & B & \\
0 & & &
\end{array}\right)
$

with $ E$ the $ (n-1) \times (n-1)$ unit matrix and
$ B=A(2:n,2:n)-u A(1,2:n)$.
Hence, the remaining eigenvalues of $ A$ can be determined by computing the eigenvalues of the $ (n-1) \times (n-1)$ matrix $ B$.

If all eigenvectors $ v$ to $ \lambda$ have zero as a first component, the construction has to be modified. In this case, we apply the deflation process to

$\displaystyle \tilde{A} = P A P\,,
$

where $ P=P^{-1}$ is a permutation matrix interchanging a non-zero element of $ v$ with the zero entry in position 1.

(Authors: Höllig/Pfeil/Walter)

see also:


[Examples]

  automatically generated 4/24/2007