On the similarity over the ring of integers of certain nilpotent matrices of maximal rank

This paper is devoted to the problem of matrix similarity recognition over the ring of integers for some families of matrices. Namely, nilpotent upper triangular matrices of maximal rank are considered such that only first and second superdiagonals of these matrices are non-zero. Several necessary conditions are obtained for similarity of such matrices to matrices of the form $\mathrm{superdiag}(a_1,a_2,\ldots,a_{n-1})$ with a single non-zero superdiagonal, that is a generalization of the Jordan cell $J_n(0)=\mathrm{superdiag}(1,1,\ldots,1)$. These conditions are formulated in simple terms of divisibility and greatest common divisors of matrix elements. The result is obtained by reducing the problem of similarity recognition to the problem of solving in integers a system of linear equations and applying the known necessary similarity conditions for arbitrary matrices. Under some additional conditions on the elements $a_1,a_2,\ldots,a_{n-1}$ of the first superdiagonal of matrix $A$, it is proven that the matrix $A$ is similar to matrix $\mathrm{superdiag}(a_1,a_2,\ldots,a_{n-1})$ regardless of the values of the elements of the second superdiagonal. Moreover, for the considered matrices of the third and the fourth orders, easily verifiable necessary and sufficient similarity conditions are obtained describing their similarity to a matrix of the form $\mathrm{superdiag}(a_1,a_2,\ldots,a_{n-1})$.