On the powers of companion matrices via linear recursiveness. Application to power of matrices

Résumé

This paper concerns some explicit formulas for the entries of the powers of companion matrices and diagonal matrices with companion matrix blocks. The development of such formulas is based on the connection of the entries of the powers of the companion matrices and the recursive sequences of Fibonacci type. Indeed, their associated fundamental Fibonacci system and fundamental sequence will play a central role. Thus, several explicit formulas for these entries are established. In addition, the Frobenus decomposition permits to apply the former results for outlining the matrix powers in the general setting. Illustrative special cases and examples are provided