Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
A rectangular matrix. Note that the orthonormal keyword is ignored in these cases.
sage: A = matrix(RDF, [[-0.978325, -0.751994, 0.925305, -0.200512, 0.420458],
... [-0.474877, -0.983403, 0.089836, 0.132218, 0.672965]])
sage: G, M = A.gram_schmidt(orthonormal=False)
sage: G.round(6).zero_at(10^-6)
[-0.607223 -0.466745 0.574315 -0.124453 0.260968]
[ 0.123203 -0.617909 -0.530578 0.289773 0.487368]
sage: M.round(6).zero_at(10^-6)
[1.611147 0.0]
[0.958116 0.867778]
sage: (A-M*G).zero_at(10^-12)
[0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0]
sage: (G*G.transpose()).round(6).zero_at(10^-6)
[1.0 0.0]
[0.0 1.0]
Even though a set of vectors may be linearly dependent, no effort is made to decide when a zero vector is
really the result of a relation of linear dependence. So in this regard, input matrices are treated as being of
full rank. Try one of the base rings that provide exact results if you need exact results.
sage: entries = [[1,1,2], [2,1,3], [3,1,4]]
sage: A = matrix(QQ, entries)
sage: A.rank()
2
sage: B = matrix(RDF, entries)
sage: G, M = B.gram_schmidt()
sage: G.round(6) # random signs
[-0.408248 -0.408248 -0.816497]
[ 0.707107 -0.707107 -0.0]
[ -0.57735 -0.57735 0.57735]
sage: M.round(10) # random
[-2.4494897428 0.0 0.0]
[-3.6742346142 0.7071067812 0.0]
[-4.8989794856 1.4142135624 0.0]
sage: (A - M*G).zero_at(1e-14)
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
sage: (G*G.transpose()).zero_at(1e-14)
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
Exact Rings, Orthonormalization:
To scale a vector to unit length requires taking a square root, which often takes us outside the base ring.
For the integers and the rationals, the field of algebraic numbers (QQbar) is big enough to contain what
we need, but the price is that the computations are very slow, hence mostly of value for small cases or
instruction. Now we need to use the orthonormal keyword.
sage: A = matrix(QQbar, [[6, -8, 1],
... [4, 1, 3],
... [6, 3, 3],
... [7, 1, -5],
... [7, -3, 5]])
sage: G, M = A.gram_schmidt(orthonormal=True)
sage: G
[ 0.5970223141259934 -0.7960297521679913 0.09950371902099891]
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