Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
(0, 1, 2, 3)
sage: B = matrix(QQ, 4, 4, entries)
sage: perm, _ = B.LU(format=’compact’); perm
(2, 0, 1, 3)
The U matrix is only guaranteed to be upper-triangular. The rows are not necessarily linearly independent,
nor are the pivots columns or rank in evidence.
sage: A = matrix(QQ, [[ 1, -4, 1, 0, -2, 1, 3, 3, 2],
... [-1, 4, 0, -4, 0, -4, 5, -7, -7],
... [ 0, 0, 1, -4, -1, -3, 6, -5, -6],
... [-2, 8, -1, -4, 2, -4, 1, -8, -7],
... [ 1, -4, 2, -4, -3, 2, 5, 6, 4]])
sage: P, L, U = A.LU()
sage: U
[ -2 8 -1 -4 2 -4 1 -8 -7]
[ 0 0 1/2 -2 -1 -2 9/2 -3 -7/2]
[ 0 0 3/2 -6 -2 0 11/2 2 1/2]
[ 0 0 0 0 -1/3 -1 5/3 -5/3 -5/3]
[ 0 0 0 0 1/3 -3 7/3 -19/3 -19/3]
sage: A.rref()
[ 1 -4 0 4 0 0 -1 -1 -1]
[ 0 0 1 -4 0 0 1 0 -1]
[ 0 0 0 0 1 0 -2 -1 -1]
[ 0 0 0 0 0 1 -1 2 2]
[ 0 0 0 0 0 0 0 0 0]
sage: A.pivots()
(0, 2, 4, 5)
TESTS:
Unknown keywords are caught.
sage: A = matrix(ZZ, 2, range(4))
sage: A.LU(pivot=’junk’)
Traceback (most recent call last):
...
ValueError: pivot strategy must be None, ’partial’ or ’nonzero’, not junk
sage: A.LU(format=’garbage’)
Traceback (most recent call last):
...
ValueError: format must be ’plu’ or ’compact’, not garbage
Components of the ‘compact’ format are immutable, while components of the ‘plu’ format are not.
sage: A = matrix(ZZ, 2, range(4))
sage: perm, M = A.LU(format=’compact’)
sage: perm[0] = 25
Traceback (most recent call last):
...
TypeError: ’tuple’ object does not support item assignment
sage: M.is_immutable()
True
sage: P, L, U = A.LU(format=’plu’)
sage: all([A.is_mutable() for A in [P, L, U]])
True
Partial pivoting is based on the absolute values of entries of a column. Trac #12208 shows that the return
value of the absolute value must be handled carefully. This tests that situation in the case of cylotomic
fields.
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