P-canonical forms and Drazin inverses

Abstract

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.

22 pages, Abstract changed, Added one reference, Some notations changed, some results added

References in corpus (2)

• A procedure for finding the k-th power of a matrix
• A closed-form expression for the kth power of semicirculant and r-circulant matrices

Cited by in corpus (3)

• Some algebraic results concerning linear recurrence sequences
• An explicit expression for the minimal polynomial of the Kronecker product of matrices. Explicit formulas for matrix logarithm and matrix exponential
• Some Explicit Formulas for Matrix Exponential, Matrix Logarithm, the $n$th Power of Matrices and their Drazin Inverses