Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[ 0 0 0 0 0 1], False)
]
TESTS:
sage: t = matrix(QQ, 3, [3, 0, -2, 0, -2, 0, 0, 0, 0]);
sage: t.decomposition_of_subspace(v, check_restrict = False) == t.decomposition_of_subspace(
True
denominator()
Return the least common multiple of the denominators of the elements of self.
If there is no denominator function for the base field, or no LCM function for the denominators, raise a
TypeError.
EXAMPLES:
sage: A = MatrixSpace(QQ,2)([’1/2’, ’1/3’, ’1/5’, ’1/7’])
sage: A.denominator()
210
A trivial example:
sage: A = matrix(QQ, 0,2)
sage: A.denominator()
1
Denominators are not defined for real numbers:
sage: A = MatrixSpace(RealField(),2)([1,2,3,4])
sage: A.denominator()
Traceback (most recent call last):
...
TypeError: denominator not defined for elements of the base ring
We can even compute the denominator of matrix over the fraction field of Z[x].
sage: K. = Frac(ZZ[’x’])
sage: A = MatrixSpace(K,2)([1/x, 2/(x+1), 1, 5/(x^3)])
sage: A.denominator()
x^4 + x^3
Here’s an example involving a cyclotomic field:
sage: K. = CyclotomicField(3)
sage: M = MatrixSpace(K,3,sparse=True)
sage: A = M([(1+z)/3,(2+z)/3,z/3,1,1+z,-2,1,5,-1+z])
sage: print A
[1/3*z + 1/3 1/3*z + 2/3 1/3*z]
[ 1 z + 1 -2]
[ 1 5 z - 1]
sage: print A.denominator()
3
density()
Return the density of the matrix.
By density we understand the ratio of the number of nonzero positions and the self.nrows() * self.ncols(),
i.e. the number of possible nonzero positions.
EXAMPLE:
149