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Deflation

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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)
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  automatically generated 4/24/2007