Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
Free module of degree 7 and rank 2 over Integer Ring
User basis matrix:
[-2 1 -3 -1 0 -1 -1]
[ 5 3 -2 5 -1 1 1]
Besides the integers, rings may be as general as principal ideal domains. Results are then free modules.
sage: R. = QQ[]
sage: A = matrix(R, [[ 1, y, 1+y^2],
... [y^3, y^2, 2*y^3]])
sage: K = A.right_kernel(algorithm=’default’, basis=’echelon’); K
Free module of degree 3 and rank 1 over Univariate Polynomial Ring in y over Rational Field
Echelon basis matrix:
[-1 -y 1]
sage: A*K.basis_matrix().transpose() == zero_matrix(ZZ, 2, 1)
True
It is possible to compute a kernel for a matrix over an integral domain which is not a PID, but usually this
will fail.
sage: D. = ZZ[]
sage: A = matrix(D, 2, 2, [[x^2 - x, -x + 5],
... [x^2 - 8, -x + 2]])
sage: A.right_kernel()
Traceback (most recent call last):
...
ArithmeticError: Ideal Ideal (x^2 - x, x^2 - 8) of Univariate Polynomial Ring in x over Inte
Matrices over non-commutative rings are not a good idea either. These are the “usual” quaternions.
sage: Q. = QuaternionAlgebra(-1,-1)
sage: A = matrix(Q, 2, [i,j,-1,k])
sage: A.right_kernel()
Traceback (most recent call last):
...
NotImplementedError: Cannot compute a matrix kernel over Quaternion Algebra (-1, -1) with ba
Sparse matrices, over the rationals and the integers, use the same routines as the dense versions.
sage: A = matrix(ZZ, [[0, -1, 1, 1, 2],
... [1, -2, 0, 1, 3],
... [-1, 2, 0, -1, -3]],
... sparse=True)
sage: A.right_kernel()
Free module of degree 5 and rank 3 over Integer Ring
Echelon basis matrix:
[ 1 0 0 2 -1]
[ 0 1 0 -1 1]
[ 0 0 1 -3 1]
sage: B = A.change_ring(QQ)
sage: B.is_sparse()
True
sage: B.right_kernel()
Vector space of degree 5 and dimension 3 over Rational Field
Basis matrix:
[ 1 0 0 2 -1]
[ 0 1 0 -1 1]
[ 0 0 1 -3 1]
With no columns, the kernel can only have dimension zero. With no rows, every possible vector is in the
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