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Let $A\in M_n(C)$. I want to show that it is similar to a Jordan matrix

By Schur's generalization,

if $A\in M_n(C)$ with $\lambda_1,\ldots\lambda_k$ distinct eigenvalues, each of multiplicity $m_i$, then $\exists S$ invertible such that

$$S^{-1}AS =\oplus_{i=1}^{k} T_{m_i}(\lambda_i) \;\text{where }\; T_{m_i}(\lambda_i) \; \text{is an } m_i \times m_i \; \text{upper triangular matrix with all its diagonal matrix equal to} \; \lambda_i $$

$$ \text{Note that} \;T_{m_i}(\lambda_i) - \lambda_i I_{m_i} \; \text{is a strictly upper triangular matrix and hence similar to a direct sum of Jordan blocks, hence: }$$

$$\exists S_i \; \text{invertible such that } T_{m_i}(\lambda_i) - \lambda_i I_{m_i}=(S_i)\big( \oplus_{j=1}^{k_i} J_{m_{i_j}}(0)\big)(S_i)^{-1}$$

Now how can I get my similarity matrix using $S$ and $S_i$

$$S^{-1}AS =\oplus_{i=1}^{k} T_{m_i}(\lambda_i)= \oplus_{i=1}^{k}\big[ (S_i)\big( \oplus_{j=1}^{k_i} J_{m_{i_j}}(0)\big)(S_i)^{-1}\big]$$

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