I am currently intensively reading my linear algebra notes under dim light and was wondering whether it is true, that a an endomorphism whose minimal polynomial has the same degree as the dimension of the vector space is similar to a companion matrix?
2026-03-25 14:37:38.1774449458
Matrix similar to a companion matrix
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The answer is yes and we have this result:
If $A$ is an $n$-by-$n$ matrix with entries from some field $K$, then the following statements are equivalent:
$A$ is similar to the companion matrix over $K$ of its characteristic polynomial
the characteristic polynomial of $A$ coincides with the minimal polynomial of $A$, equivalently the minimal polynomial has degree $n$
there exists a cyclic vector $v$ in $V=K^n$ for $A$, meaning that $\{v, Av, A_2v,\ldots, A_{n−1}v\}$ is a basis of $V$.
Source http://en.wikipedia.org/wiki/Companion_matrix