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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) \,. $

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  automatically generated 5/17/2011