On the similarity of upper triangular nilpotent matrices of the $4$th and the $5$th orders to a generalized Jordan block over the ring of integers

In this paper conditions for similarity of an upper triangular nilpotent matrix and a generalized Jordan block (i. e. a matrix where only the elements of the first superdiagonal are non-zero) are considered. The problem is solved over the ring of integers. Necessary and sufficient conditions for similarity to a generalized Jordan block are obtained for the following classes of matrices: the fourth-order matrices of rank $3$ with nonzero elements of the first superdiagonal; the fifth-order matrices of rank $4$ and some additional restrictions on the elements of the first superdiagonal. These conditions are formulated in simple terms of divisibility and greatest common divisors of matrix elements. It is proved that if the first and last elements of the first superdiagonal are coprime, and the product of the remaining elements of this superdiagonal is equal to $1,$ then this matrix is similar to a generalized Jordan block. To obtain the similarity criterion, the following statement is used: if two nilpotent upper triangular matrices of order $n$ and rank $n - 1$ are similar over the ring of integers, then among the transforming matrices there is a triangular matrix. This statement reduces the problem of recognizing similarity to solving a system of linear equations in integers. The main tool for obtaining the results in the article is the criterion of consistency of a system of linear equations over the ring of integers.