A matrix in Macaulay is a matrix of polynomials over some polynomial ring. In this chapter, we describe the operations available for computing with matrices which don't require standard bases. These range from matrix input and output, to simple operations such as addition and multiplication of matrices, to relatively esoteric operations such as koszul or contract .
Matrices have several interpretations in Macaulay .
Let M be an m by n matrix of polynomials over the ring R . There are two R -modules which we can associate to M . First, the submodule is the submodule of generated by the columns of the matrix M . The second module corresponding to M is the module N which is presented by M . That is, The R -module N has m generators, one for each row of M , and the relations between these generators are exactly the columns of the matrix M .