Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: t.column_module()
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 -1]
[ 0 1 2]
column_space()
Return the vector space over the base ring spanned by the columns of this matrix.
EXAMPLES:
sage: M = MatrixSpace(QQ,3,3)
sage: A = M([1,9,-7,4/5,4,3,6,4,3])
sage: A.column_space()
Vector space of degree 3 and dimension 3 over Rational Field
Basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]
sage: W = MatrixSpace(CC,2,2)
sage: B = W([1, 2+3*I,4+5*I,9]); B
[ 1.00000000000000 2.00000000000000 + 3.00000000000000*I]
[4.00000000000000 + 5.00000000000000*I 9.00000000000000]
sage: B.column_space()
Vector space of degree 2 and dimension 2 over Complex Field with 53 bits of precision
Basis matrix:
[ 1.00000000000000 0.000000000000000]
[0.000000000000000 1.00000000000000]
conjugate()
Return the conjugate of self, i.e. the matrix whose entries are the conjugates of the entries of self.
EXAMPLES:
sage: A = matrix(CDF, [[1+I,1],[0,2*I]])
sage: A.conjugate()
[1.0 - 1.0*I 1.0]
[ 0.0 -2.0*I]
A matrix over a not-totally-real number field:
sage: K. = NumberField(x^2+5)
sage: M = matrix(K, [[1+j,1], [0,2*j]])
sage: M.conjugate()
[-j + 1 1]
[ 0 -2*j]
There is a shortcut for the conjugate:
sage: M.C
[-j + 1 1]
[ 0 -2*j]
There is also a shortcut for the conjugate transpose, or “Hermitian transpose”:
sage: M.H
[-j + 1 0]
[ 1 -2*j]
Conjugates work (trivially) for matrices over rings that embed canonically into the real numbers:
142 Chapter 7. Base class for matrices, part 2