Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[0 0 1 0 0]
[0 1 0 0 0]
sage: L
[ 1 0 0 0 0]
[ 1/2 1 0 0 0]
[ 1/2 1/3 1 0 0]
[ 1 2/3 1/5 1 0]
[ 1/2 -1/3 -2/5 0 1]
sage: U
[ 2 -1 0 6 4 8 -2]
[ 0 3/2 2 -5 -3 -3 4]
[ 0 0 -5/3 20/3 -2 -4 -10/3]
[ 0 0 0 0 2/5 4/5 0]
[ 0 0 0 0 1/5 2/5 0]
sage: A == P*L*U
True
sage: P, L, U = A.LU(pivot=’nonzero’)
sage: P
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
sage: L
[ 1 0 0 0 0]
[ 2 1 0 0 0]
[ 2 2 1 0 0]
[ 1 1 -1 1 0]
[ 1 2 2 0 1]
sage: U
[ 1 -1 0 2 4 7 -1]
[ 0 1 0 2 -4 -6 0]
[ 0 0 1 -4 2 4 2]
[ 0 0 0 0 1 2 0]
[ 0 0 0 0 -1 -2 0]
sage: A == P*L*U
True
An example of the compact format.
sage: B = matrix(QQ, [[ 1, 3, 5, 5],
... [ 1, 4, 7, 8],
... [-1, -4, -6, -6],
... [ 0, -2, -5, -8],
... [-2, -6, -6, -2]])
sage: perm, M = B.LU(format=’compact’)
sage: perm
(4, 3, 0, 1, 2)
sage: M
[ -2 -6 -6 -2]
[ 0 -2 -5 -8]
[-1/2 0 2 4]
[-1/2 -1/2 3/4 0]
[ 1/2 1/2 -1/4 0]
We can easily illustrate the relationships between the two formats with a square matrix.
sage: C = matrix(QQ, [[-2, 3, -2, -5],
... [ 1, -2, 1, 3],
122 Chapter 7. Base class for matrices, part 2