Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: m = MatrixSpace(ZZ, 2,0)(0); d,u,v = m.smith_form(); u*m*v == d
True
sage: m = MatrixSpace(ZZ, 0,2)(0); d,u,v = m.smith_form(); u*m*v == d
True
sage: m = MatrixSpace(ZZ, 0,0)(0); d,u,v = m.smith_form(); u*m*v == d
True
See Also:
elementary_divisors()
stack(bottom, subdivide=False)
Return the matrix self on top of bottom: [ self ] [ bottom ]
EXAMPLES:
sage: M = Matrix(ZZ, 2, 3, range(6))
sage: N = Matrix(ZZ, 1, 3, [10,11,12])
sage: M.stack(N)
[ 0 1 2]
[ 3 4 5]
[10 11 12]
A vector may be stacked below a matrix.
sage: A = matrix(ZZ, 2, 4, range(8))
sage: v = vector(ZZ, 4, range(4))
sage: A.stack(v)
[0 1 2 3]
[4 5 6 7]
[0 1 2 3]
The subdivide option will add a natural subdivision between self and
bottom. For more details about how subdivisions are managed when stacking, see
sage.matrix.matrix1.Matrix.stack().
sage: A = matrix(ZZ, 3, 4, range(12))
sage: B = matrix(ZZ, 2, 4, range(8))
sage: A.stack(B, subdivide=True)
[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[-----------]
[ 0 1 2 3]
[ 4 5 6 7]
TESTS:
Stacking a dense matrix atop a sparse one should work:
sage: M = Matrix(ZZ, 2, 3, range(6))
sage: M.is_sparse()
False
sage: N = diagonal_matrix([10,11,12], sparse=True)
sage: N.is_sparse()
True
sage: P = M.stack(N); P
[ 0 1 2]
[ 3 4 5]
[10 0 0]
[ 0 11 0]
335