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Let $J=\operatorname{diag}\left( J\left( \mu_{1}\right) ,\ldots,J\left( \mu_{r}\right) \right) $ be a Jordan matrix of dimension $n\times n$ with Jordan blocks

$$J( \mu_{i}) = \left( \begin{array} [c]{cccc}% \mu_{i} & 1 & 0 & 0\\ 0 & \mu_{i} & \ddots & 0\\ \vdots & \vdots & \ddots & 1\\ 0 & 0 & \cdots & \mu_{i}% \end{array} \right) \in \mathbb{C}^{n_{i}\times n_{i}}$$

Let $\lambda_{1},\ldots,\lambda_{r}\in\mathbb{C}$ such that $\lambda _{1},\ldots,\lambda_{r}\notin\sigma\left( J\right) $, (where $\sigma\left( J\right) $ stands for the spectrum of $J$). Let define the vector $w=e_{n_{1}}+e_{n_{1}+n_{2}}+\cdots+e_{n_{1}+n_{2}+\cdots+n_{r}}$ where $e_{i}$ represents a vector of the standard basis with entry $1$ at position $i$. (In fact $e_{n_{1}+n_{2}+\cdots+n_{k}}$ is a generator of the cyclic subspace associated with $J\left( \mu_{k}\right) $).

Let define the set $\mathcal{S}$: \begin{align*} \mathcal{S} & =\bigcup\limits_{i=1}^{r}\mathcal{S}_{i}\\ \mathcal{S}_{i} & =\left\{ \left( J-\lambda_{i}I\right) ^{-j}% w:j=1,\ldots,n_{i}\right\} \end{align*} Prove that $\mathcal{S}_{0}\subset\mathcal{S}$ such that $\operatorname{card}\left( \mathcal{S}_{0}\right) =n$ is a linearly independent set.

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  • $\begingroup$ Presumably, $n_k = \dim J(\mu_k)$ for each $k=1, \ldots, r$? $\endgroup$ – Sammy Black Oct 20 '13 at 9:31
  • $\begingroup$ Never mind. I see it in your definition of $J(\mu_i)$. $\endgroup$ – Sammy Black Oct 20 '13 at 9:32

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