[math] \newcommand{\mathds}{\mathbb}[/math]

Prove that any matrix can be put in Jordan form,

[[math]] A\sim\begin{pmatrix} J_1\\ &\ddots\\ &&J_k \end{pmatrix} [[/math]]

with each of the blocks which appear, called Jordan blocks, being as follows,

[[math]] J_i=\begin{pmatrix} \lambda_i&1\\ &\lambda_i&1\\ &&\ddots&\ddots\\ &&&\lambda_i&1\\ &&&&\lambda_i \end{pmatrix} [[/math]]

with the size being the multiplicity of [math]\lambda_i[/math].