P-canonical forms and Drazin inverses

In this paper, $\mathcal{P}$-canonical forms of $(A^{k})_{k}$ (or simply of the matrix $A$) are defined and some of their properties are proved. It is also shown how we can deduce from them many interesting informations about the matrix $A$. In addition, it is proved that the $\mathcal{P}$-canonical forms of $A$ can be written as a sum of two parts, the geometric and the non-geometric parts of $A$, and that the $\mathcal{P}$-canonical form of the Drazin inverse $A_{d}$ of $A$ can be deduced by simply plugging $-k$ for $k$ in the geometric part of $A$. Finally, several examples are provided to illustrate the obtained results.