Extremal matrix centralizers (Q1940090)

The centralizer \(C(A)\) of an \(n\times n\) matrix \(A\) over some field comprises the set of all matrices that commute with \(A\). A matrix \(A\) is called minimal if \(C(A) \supseteq C(X) \Rightarrow C(A) = C(X)\), and maximal if \(C(A)\subseteq C(X) \Rightarrow C(A) = C(X)\). It is known that a matrix \(A\) over \(\mathbb C\) is minimal if and only if it is nonderogatory (minimal polynomial equals characteristic polynomial). A necessary and sufficient condition for a matrix \(A\) over \(\mathbb C\) to be maximal has also been found. In this paper the authors extend these characterizations to matrices over an arbitrary field with sufficiently many elements.