Commutative matrix subalgebras generated by nonderogatory matrices

The paper is devoted to studying the question on the number of subalgebras in the matrix algebra that generated by nonderogatory matrices and are distinct up to similarity. A criterion for similarity of matrix algebras of the type indicated is established in terms of isomorphisms of quotient algebras of the algebra of polynomials. The existence of infinite fields for which the number of distinct algebras is infinite for all values of the matrix order is established. For algebraically closed fields, for the field of real numbers, and for finite fields of sufficiently large cardinality, the number of distinct algebras generated by nonderogatory matrices is determined as a function of the matrix order.