Internal format of a computation

 

When any matrix is defined in Macaulay, its internal format includes the following information.

1. An array of columns, where each column is a vector of polynomials.
2. The degree of each row.
3. The degree of each column.

This information can be obtained by using the type, row-degs, and col-degs commands.

After a computation is performed using one of the computation commands, each matrix which has had a standard basis computed also contains the following information.

1. The standard basis, represented by an array of columns
2. The change of basis matrix, C, represented by an array of columns. The column is the representation of the standard basis element in terms of the generators of the matrix. Thus, if there are s elements in the standard basis, and r columns of the original matrix M, then C is a r by s matrix.

Not all computation commands compute a change of basis matrix. The commands which do include lift-std, res, and nres. The other computation commands do not compute this matrix. Most commands of Macaulay do not require the change of basis matrix C. The only exceptions are the reduce and lift commands. The commands res and nres compute C since this is used to calculate syzygy modules.

The following two commands either display or create a new matrix consisting of the standard basis or change of basis matrix.