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Cyclic Bases of Generalized Eigenspaces


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A generalised eigenspace of a matrix $ A$ can be decomposed in a direct sum of cyclic subspaces:

$\displaystyle H_\lambda = V_1 \oplus \cdots \oplus V_\ell\,,
$

that is, each subspace $ V_i$ has a basis of the form

$\displaystyle B^{k_i}v_i,\,\ldots,\,Bv_i,\,v_i,
\quad B=A-\lambda E,
$

where $ w_i = B^{k_i}v_i$ is a eigenvector for eigenvalue $ \lambda$.

The subspaces $ V_i$ are invariant under matrix $ A$. The restriction of $ A$ onto these subspaces has the representation:

$\displaystyle J_i =
\left(\begin{array}{cccc}
\lambda & 1 & & 0 \\
& \lambda & \ddots \\
& & \ddots & 1 \\
0 & & & \lambda
\end{array}\right)
\,.
$

see also:


  automatically generated 5/17/2011