CMR  1.3.0
(Strongly) k-Modular and Unimodular Matrices

Consider a matrix \( M \in \mathbb{Z}^{m \times n} \) of rank \( r \). The matrix \( M \) is called k-modular (for some \( k \in \{1,2,\dotsc \} \)) if these two conditions are satisfied:

In case \( M \) has full row-rank, the first property requires that the determinant of any basis matrix shall be \( k \), while the second property requires that \( M_{\star,B}^{-1} M \) is totally unimodular. Otherwise, \( M_{\star,B} \) is singular, and hence the property is more technical.

Note
k-modularity is independent of the choice of the column basis \( B \).

Additionally, \( M \) is called strongly k-modular if \( M \) and \( M^{\textsf{T}} \) are k-modular. The special cases with \( k = 1 \) is called unimodular and strongly unimodular, respectively.

Usage

The executable cmr-k-modular determines whether a given matrix \( M \) is \(k\)-modular (and determines \( k \)).

./cmr-k-modular [OPTION]... FILE

Options:

Formats for matrices are dense-matrix and sparse-matrix. If FILE is -, then the input will be read from stdin.

C Interface

The functionality is defined in k_modular.h. The main functions are: