Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
... [-4, 7, -3, -8],
... [-3, 8, -1, -5]])
sage: P, L, U = C.LU(format=’plu’)
sage: perm, M = C.LU(format=’compact’)
sage: (L - identity_matrix(4)) + U == M
True
sage: p = [perm[i]+1 for i in range(len(perm))]
sage: PP = Permutation(p).to_matrix()
sage: PP == P
True
For a nonsingular matrix, and the ‘nonzero’ pivot strategy there is no need to permute rows, so the permutation
matrix will be the identity. Furthermore, it can be shown that then the L and U matrices are uniquely
determined by requiring L to have ones on the diagonal.
sage: D = matrix(QQ, [[ 1, 0, 2, 0, -2, -1],
... [ 3, -2, 3, -1, 0, 6],
... [-4, 2, -3, 1, -1, -8],
... [-2, 2, -3, 2, 1, 0],
... [ 0, -1, -1, 0, 2, 5],
... [-1, 2, -4, -1, 5, -3]])
sage: P, L, U = D.LU(pivot=’nonzero’)
sage: P
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
sage: L
[ 1 0 0 0 0 0]
[ 3 1 0 0 0 0]
[ -4 -1 1 0 0 0]
[ -2 -1 -1 1 0 0]
[ 0 1/2 1/4 1/2 1 0]
[ -1 -1 -5/2 -2 -6 1]
sage: U
[ 1 0 2 0 -2 -1]
[ 0 -2 -3 -1 6 9]
[ 0 0 2 0 -3 -3]
[ 0 0 0 1 0 4]
[ 0 0 0 0 -1/4 -3/4]
[ 0 0 0 0 0 1]
sage: D == L*U
True
The base ring of the matrix may be any field, or a ring which has a fraction field implemented in Sage. The
ring needs to be exact (there is a numerical LU decomposition for matrices over RDF and CDF). Matrices
returned are over the original field, or the fraction field of the ring. If the field is not ordered (i.e. the
absolute value function is not implemented), then the pivot strategy needs to be ‘nonzero’.
sage: A = matrix(RealField(100), 3, 3, range(9))
sage: P, L, U = A.LU()
Traceback (most recent call last):
...
TypeError: base ring of the matrix must be exact, not Real Field with 100 bits of precision
sage: A = matrix(Integers(6), 3, 2, range(6))
sage: A.LU()
123