CMR  1.3.0
(Strongly) Equimodular and Unimodular Matrices

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 IN-MAT [OPTION]...

Options:

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

Advanced options:

  • --stats Print statistics about the computation to stderr.
  • --time-limit LIMIT Allow at most LIMIT seconds for the computation.

Formats for matrices: dense, sparse If IN-MAT is -, then the input will be read from stdin.

C Interface

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

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