Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: from sage.matrix.constructor import random_subspaces_matrix
sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 6, 8)
sage: B=random_subspaces_matrix(matrix_space, rank=3); B # random
[ 113 339 -46 218 -243 641 -269 -306]
[ -33 -99 13 -63 69 -185 77 90]
[ 35 105 -14 67 -74 197 -82 -95]
[ -18 -54 7 -34 37 -100 41 49]
[ -26 -78 10 -49 53 -144 59 71]
[ -8 -24 3 -15 16 -44 18 22]
sage: B.rank()
3
sage: B.nullity()
3
sage: (B.nrows(), B.ncols())
(6, 8)
sage: all([x in ZZ for x in B.list()])
True
sage: B_expanded=B.augment(identity_matrix(6)).rref()
sage: all([x in ZZ for x in B_expanded.list()])
True
sage: B_expanded # random
[ 1 3 0 0 1 1 3 -2 0 0 -3 0 -9 16]
[ 0 0 1 0 3 -2 -1 -3 0 0 2 0 11 -27]
[ 0 0 0 1 -1 2 -3 -1 0 0 2 0 7 -14]
[ 0 0 0 0 0 0 0 0 1 0 -5 0 -3 2]
[ 0 0 0 0 0 0 0 0 0 1 1 0 1 -3]
[ 0 0 0 0 0 0 0 0 0 0 0 1 -1 1]
Check that we fixed Trac #10543 (echelon forms should be immutable):
sage: B_expanded.is_immutable()
True
We want to modify B_expanded, so replace it with a copy:
sage: B_expanded = copy(B_expanded)
sage: B_expanded.subdivide(B.nrows()-B.nullity(),B.ncols());B_expanded # random
[ 1 3 0 0 1 1 3 -2| 0 0 -3 0 -9 16]
[ 0 0 1 0 3 -2 -1 -3| 0 0 2 0 11 -27]
[ 0 0 0 1 -1 2 -3 -1| 0 0 2 0 7 -14]
[-------------------------------+-----------------------]
[ 0 0 0 0 0 0 0 0| 1 0 -5 0 -3 2]
[ 0 0 0 0 0 0 0 0| 0 1 1 0 1 -3]
[ 0 0 0 0 0 0 0 0| 0 0 0 1 -1 1]
sage: C=B_expanded.subdivision(0,0)
sage: C # random
[ 1 3 0 0 1 1 3 -2]
[ 0 0 1 0 3 -2 -1 -3]
[ 0 0 0 1 -1 2 -3 -1]
sage: L=B_expanded.subdivision(1,1)
sage: L # random
[ 1 0 -5 0 -3 2]
[ 0 1 1 0 1 -3]
[ 0 0 0 1 -1 1]
sage: B.right_kernel()==C.right_kernel()
True
sage: B.row_space()==C.row_space()
True
sage: B.column_space()==L.right_kernel()
54 Chapter 2. Matrix Constructor