Random matrices for practice

The topic in class this week is abstract vector spaces. Sage does understand abstract vector spaces (and many other abstract algebraic structures) to some extent, but not in a way that is useful at the level of this class. So, here’s a brief discussion of creating matrices for computational practice.

First, for row reduction practice, it’s nice to be able to create a matrix of a given size and rank, with integer entries which aren’t too huge:
random_matrix(ZZ,4,5,algorithm='echelonizable',rank=3, upper_bound=7)
will create a random 4×5 matrix with rank 3, integer entries (that’s the ZZ), and entries no bigger than 7.

Second, for inverting matrices, it’s nice if A has integer entries and the inverse of A does, too. Since det(AB)=det(A)det(B), if the determinant of A is not +/-1 then the entries of A^-1 will not be integers. The converse is also true, remarkably: if |det(A)|=1 then the entries of A^-1 will be integers. (This is one of the very few useful applications of Cramer’s rule.) So, we need a way of creating matrices with determinant 1 (or -1). Such matrices are called “unimodular”:
random_matrix(ZZ,5,algorithm='unimodular',upper_bound=9)
will create a 5×5 matrix with integer entries, determinant 1, and entries no bigger than 9 (or maybe 8) in absolute value.

Have fun…