### Nonderogatory of Sum and Product of Doubly Companion Matrices

#### Abstract

Butcher and Chartier in [1, pp. 274–276] first introduced the doubly companion matrices, after that Butcher and Wright [2] and Wright [3] used of doubly companion matrices as a tool to analyze numerical methods and some general linear methods property. In this paper, we prove that any doubly companion matrix, and the sum of two doubly companion matrices are nonderogatory, and obtain the explicit form of its minimal polynomials. Moreover, we construct some examples which show that those product of two doubly companion matrices may not be a nonderogatory matrix. As in [4], we gives some condition for which the product of (doubly) companion matrices is a nonderogatory matrix. In addition we assert that the product of two unreduced Hessenberg is not nonderogatory.

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