Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
–‘generic’ - naive algorithm usable for matrices over any field
–‘pari’ - PARI library code for matrices over number fields or the integers
–‘padic’ - padic algorithm from IML library for matrices over the rationals and integers
–‘pluq’ - PLUQ matrix factorization for matrices mod 2
•basis - default: ‘echelon’ - a keyword that describes the format of the basis returned. Allowable
values are:
OUTPUT:
–‘echelon’: the basis matrix is in echelon form
–‘pivot’ : each basis vector is computed from the reduced row-echelon form of self by placing a
single one in a non-pivot column and zeros in the remaining non-pivot columns. Only available
for matrices over fields.
–‘LLL’: an LLL-reduced basis. Only available for matrices over the integers.
–‘computed’: no work is done to transform the basis, it is returned exactly as provided by
whichever routine actually computed the basis. Request this for the least possible computation
possible, but with no guarantees about the format of the basis.
A matrix X whose rows are an independent set spanning the right kernel of self.
self*X.transpose() is a zero matrix.
The output varies depending on the choice of algorithm and the format chosen by basis.
The results of this routine are not cached, so you can call it again with different options to get possibly
different output (like the basis format). Conversely, repeated calls on the same matrix will always start
from scratch.
Note: If you want to get the most basic description of a kernel, with a minimum of overhead, then ask for
the right kernel matrix with the basis format requested as ‘computed’. You are then free to work with the
output for whatever purpose. For a left kernel, call this method on the transpose of your matrix.
For greater convenience, plus cached results, request an actual vector space or free module with
right_kernel() or left_kernel().
EXAMPLES:
Over the Rational Numbers:
Kernels are computed by the IML library in _right_kernel_matrix(). Setting the algorithm
keyword to ‘default’, ‘padic’ or unspecified will yield the same result, as there is no optional behavior.
The ‘computed’ format of the basis vectors are exactly the negatives of the vectors in the ‘pivot’ format.
sage: A = matrix(QQ, [[1, 0, 1, -3, 1],
... [-5, 1, 0, 7, -3],
... [0, -1, -4, 6, -2],
... [4, -1, 0, -6, 2]])
sage: C = A.right_kernel_matrix(algorithm=’default’, basis=’computed’); C
[-1 2 -2 -1 0]
[ 1 2 0 0 -1]
sage: A*C.transpose() == zero_matrix(QQ, 4, 2)
True
sage: P = A.right_kernel_matrix(algorithm=’padic’, basis=’pivot’); P
[ 1 -2 2 1 0]
[-1 -2 0 0 1]
sage: A*P.transpose() == zero_matrix(QQ, 4, 2)
So
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