Jordan

The Jordan decomposition of a matrix $A$ is a change of basis where $A$ is written into a diagonal or quasi-diagonal form: $A=P^{-1}JP$, where $P$ is the change of basis matrix and $J$ is a matrix with the following structure

$\left(\begin{array}{ccccccccc}{\lambda }_1& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& {\lambda }_1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& {\lambda }_1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {\lambda }_2& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& {\lambda }_2& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& {\lambda }_3& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \ddots & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {\lambda }_n& 1\\ 0& 0& 0& 0& 0& 0& 0& 1& {\lambda }_n\end{array}\right)$

where ${\lambda }_1,...,{\lambda }_n$ are the eigenvalues of $A$.

jordan(Matrix)

Description

Given a matrix, returns the matrix $J$. If the option transformation_matrix is set to true, the output is a list with $J$ and $P$.