Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
True
sage: B.left_kernel()==L.row_space()
True
A matrix to show that the null space of the L matrix is the column space of the starting matrix.
sage: A=random_matrix(QQ, 5, 7, algorithm=’subspaces’, rank=None); A # random
[-31 12 -9 -27 21 2 -15]
[105 -24 6 103 -30 -34 79]
[ 29 -9 5 26 -14 -5 17]
[233 -55 16 228 -71 -73 173]
[-42 10 -3 -41 13 13 -31]
sage: (A.nrows(), A.ncols())
(5, 7)
sage: all([x in ZZ for x in A.list()])
True
sage: A.nullity() # random
1
sage: A_expanded=A.augment(identity_matrix(5)).rref()
sage: A_expanded # random
[ 1 0 0 0 0 1 0 0 3 7 25 151]
[ 0 1 0 0 1 2 1 0 5 21 84 493]
[ 0 0 1 0 -1 2 0 0 2 13 53 308]
[ 0 0 0 1 0 -1 1 0 -2 -3 -9 -57]
[ 0 0 0 0 0 0 0 1 -3 1 1 -2]
sage: all([x in ZZ for x in A_expanded.list()])
True
sage: C=A_expanded.submatrix(0,0,A.nrows()-A.nullity(),A.ncols())
sage: L=A_expanded.submatrix(A.nrows()-A.nullity(),A.ncols())
sage: A.right_kernel()==C.right_kernel()
True
sage: A.row_space()==C.row_space()
True
sage: A.column_space()==L.right_kernel()
True
sage: A.left_kernel()==L.row_space()
True
TESTS:
The designated rank of the L matrix cannot be greater than the number of desired rows, nor can the rank be
negative.
sage: random_matrix(QQ, 19, 20, algorithm=’subspaces’, rank=21)
Traceback (most recent call last):
...
ValueError: rank cannot exceed the number of rows or columns.
sage: random_matrix(QQ, 19, 20, algorithm=’subspaces’, rank=-1)
Traceback (most recent call last):
...
ValueError: matrices must have rank zero or greater.
REFERENCES:
AUTHOR:
Billy Wonderly (2010-07)
55