Matroid Decomposition REVISED EDITION

Matrix We take a detour to introduce abstract matrices. We want to acquire a good understanding of such matrices, since they not only represent matroids, but 74 Chapter 3. From Graphs to Matroids also display a lot of structural information about matroids that other ways do not. An abstract matrix B is a {0, 1} matrix with row and column indices plus a function called abstract determinant and denoted by det. The function det associates with each square submatrix D of the {0, 1} matrix the value 0 or 1, i.e., detD is 0 or 1. Note that numerically identical square submatrices with differing row or column index sets may have different determinants. The reader should not be misled by the symbols 0 and 1. Indeed, for the moment, we do not view abstract matrices as part of some algebraic structure. It turns out, though, that 0 and 1 allow a rather appealing use of linear algebra terms. For example, we call D nonsingular if detD = 1, and singular otherwise. The function det must obey several conditions. First, if D is the 1× 1 matrix [ 0 ] (resp. [ 1 ]), then detD = 0 (resp. detD = 1). Second, for any nonempty submatrix B of B, the maximal nonsingular submatrices must have the same size. This condition may be rephrased as follows. Start with some nonsingular submatrix of B. Iteratively add a row and a column such that each time another nonsingular submatrix results. Stop when no further enlargement is possible. The above maximality condition demands that the order of the final nonsingular submatrix is the same regardless of the choice of the initial nonsingular submatrix and of the rows and columns added to it. The order of any such final nonsingular submatrix is called the rank of B. For the case where B is trivial or empty, we declare rank B to be 0. Upon deletion of a column or row, we demand that the rank drop at most by the rank of that row or column. Third, the rank function of B must behave much like the rank function of matrices over fields. In particular, for any partition of any submatrix of B of the form (3.4.5) B11 B12 B13 B21 B23 B22 B31 B32 B33 Partitioned submatrix of B