The matroid represented by a matrix \( M \in \mathbb{F}^{m \times n} \) has \( m + n \) elements that correspond to the columns of \( [ \mathbb{I} \mid M ] \). A subset of these columns is independent if and only if it is linearly independent over the field \( \mathbb{F} \). Applying a pivot operation (over \( \mathbb{F} \)) and exchanging the corresponding row and column elements leaves the matroid unchanged. In CMR, pivots are carried out via CMRchrmatBinaryPivot, CMRchrmatBinaryPivots, CMRchrmatTernaryPivot and CMRchrmatTernaryPivots.

Minors

A minor of a represented matroid is another one obtained by deleting or contracting elements. An element associated with a row (resp. column) can be contracted (resp. deleted) by removing the corresponding matrix row (resp. column). In order to delete a row element or contract a column element one must first pivot such that the element type (row/column) is changed.

A minor of a matroid represented by matrix \( M \) is represented by means of a CMR_MINOR object. It consists of a (potentially empty) array of pivots and a CMR_SUBMAT object indicating a submatrix \( M' \) of the matrix obtained from \( M \) after applying the pivots. Moreover, the type field indicates a certain structure of \( M' \):

A determinant type indicates that \( M' \) is a submatrix of \( M \) with \( |\det(M')| \geq 2 \). In particular, no pivots are applied.
A Fano type indicates that \( M' \) represents the Fano matroid \( F_7 \).
A Fano-dual type indicates that \( M' \) represents the dual \( F_7^\star \) of the Fano matroid.
A K5 type indicates that \( M' \) represents the graphic matroid \( M(K_5) \) of the complete graph \( K_5 \).
A K5-dual type indicates that \( M' \) represents the dual matroid \( M(K_5)^\star \) of the complete graph \( K_5 \).
A K33 type indicates that \( M' \) represents the graphic matroid \( M(K_{3,3}) \) of the complete bipartite graph \( K_{3,3} \).
A K33-dual type indicates that \( M' \) represents the dual matroid \( M(K_{3,3})^\star \) of the complete bipartite graph \( K_{3,3} \).