# Every matrix is similar to a Jordan matrix, help finding similarity matrix

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 $$S_i$$

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