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

for some column basis \( B \subseteq \{1,2,\dotsc,n\} \) of \( M \), the greatest common divisor of the determinants of all \(r \)-by- \( r \) submatrices of \( M_{\star,B} \) is equal to \( k \).
The matrix \( X \) such that \( M = M_{\star,B} X \) is totally unimodular.

In case \( M \) has full row-rank, the first property requires that the determinant of any basis matrix shall be \( \pm k \), while the second property requires that \( M_{\star,B}^{-1} M \) is totally unimodular. Otherwise, \( M_{\star,B} \) is not square, and hence the property is more technical.

Note: Equimodularity is independent of the choice of the column basis \( B \).

Additionally, \( M \) is called strongly equimodular if \( M \) and \( M^{\textsf{T}} \) are both equimodular, which implies that they are equimodular for the same gcd determinants. The special cases with \( k = 1 \) are called unimodular and strongly unimodular, respectively.

Usage

The executable cmr-equimodular determines whether a given matrix \( M \) with determinant gcd \( k \).

./cmr-equimodular [OPTION]... FILE

Options:

-i FORMAT Format of input FILE; default: dense.
-t Test \( M^{\textsf{T}} \) instead.
-s Test for strong equimodularity.
-u Test only for unimodularity, i.e., \( k = 1 \).

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 equimodular.h. The main functions are:

CMRequimodularTest() tests a matrix for being equimodular.
CMRequimodularTestStrong() tests a matrix for being strongly equimodular.
CMRunimodularTest() tests a matrix for being unimodular.
CMRunimodularTestStrong() tests a matrix for being strongly unimodular.