Miscellaneous module-related functions¶

AUTHORS:

William Stein (2007-11-18)

sage.modules.misc.gram_schmidt(B)[source]¶

Return the Gram-Schmidt orthogonalization of the entries in the list B of vectors, along with the matrix mu of Gram-Schmidt coefficients.

Note that the output vectors need not have unit length. We do this to avoid having to extract square roots.

Note

Use of this function is discouraged. It fails on linearly dependent input and its output format is not as natural as it could be. Instead, see sage.matrix.matrix2.Matrix2.gram_schmidt() which is safer and more general-purpose.

EXAMPLES:

Sage

sage: B = [vector([1,2,1/5]), vector([1,2,3]), vector([-1,0,0])]
sage: from sage.modules.misc import gram_schmidt
sage: G, mu = gram_schmidt(B)
sage: G
[(1, 2, 1/5), (-1/9, -2/9, 25/9), (-4/5, 2/5, 0)]
sage: G[0] * G[1]
0
sage: G[0] * G[2]
0
sage: G[1] * G[2]
0
sage: mu
[      0       0       0]
[   10/9       0       0]
[-25/126    1/70       0]
sage: a = matrix([])
sage: a.gram_schmidt()
([], [])
sage: a = matrix([[],[],[],[]])
sage: a.gram_schmidt()
 ([], [])

Python

>>> from sage.all import *
>>> B = [vector([Integer(1),Integer(2),Integer(1)/Integer(5)]), vector([Integer(1),Integer(2),Integer(3)]), vector([-Integer(1),Integer(0),Integer(0)])]
>>> from sage.modules.misc import gram_schmidt
>>> G, mu = gram_schmidt(B)
>>> G
[(1, 2, 1/5), (-1/9, -2/9, 25/9), (-4/5, 2/5, 0)]
>>> G[Integer(0)] * G[Integer(1)]
0
>>> G[Integer(0)] * G[Integer(2)]
0
>>> G[Integer(1)] * G[Integer(2)]
0
>>> mu
[      0       0       0]
[   10/9       0       0]
[-25/126    1/70       0]
>>> a = matrix([])
>>> a.gram_schmidt()
([], [])
>>> a = matrix([[],[],[],[]])
>>> a.gram_schmidt()
 ([], [])

Linearly dependent input leads to a zero dot product in a denominator. This shows that Issue #10791 is fixed.

Sage

sage: from sage.modules.misc import gram_schmidt
sage: V = [vector(ZZ,[1,1]), vector(ZZ,[2,2]), vector(ZZ,[1,2])]
sage: gram_schmidt(V)
Traceback (most recent call last):
...
ValueError: linearly dependent input for module version of Gram-Schmidt

Python

>>> from sage.all import *
>>> from sage.modules.misc import gram_schmidt
>>> V = [vector(ZZ,[Integer(1),Integer(1)]), vector(ZZ,[Integer(2),Integer(2)]), vector(ZZ,[Integer(1),Integer(2)])]
>>> gram_schmidt(V)
Traceback (most recent call last):
...
ValueError: linearly dependent input for module version of Gram-Schmidt