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orthogonalization and orthonormalization of vectors under integer mod

asked 2022-01-30 13:44:45 +0200

osi gravatar image

how can I get first orthogonal(Gram–Schmidt process) and then orthonormal of basis vector list under integer mod.

l=[(1,2,3),(1,5,6)]
basis=[vector(GF(29),i) for i in l]
print(basis)
#[(1,2,3),(1,5,6)]
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answered 2022-01-30 15:48:35 +0200

tmonteil gravatar image

updated 2022-01-30 15:50:44 +0200

Here is a partial answer : instead of putting your "basis" in a list, you should put it in a matrix, so that you could benefit from the methods available there.

sage: M = matrix(basis)
sage: M
[1 2 3]
[1 5 6]
sage: M.parent()
Full MatrixSpace of 2 by 3 dense matrices over Finite Field of size 29
sage: M.gram_schmidt()
AttributeError: 'sage.rings.finite_rings.integer_mod.IntegerMod_int' object has no attribute 'conjugate'

Unfortunately, to compute the Gram-Schmidt orthogonalization of M, Sage needs to find the conjugate of elements GF(29), which seems not defined (if you replace GF(29) with ZZ, everythin seems fine).

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answered 2022-01-30 18:16:37 +0200

Max Alekseyev gravatar image

You can perform Gram–Schmidt over QQ and then change ring to GF(29):

M = Matrix( QQ, [(1,2,3),(1,5,6)] )
[ t.change_ring(GF(29)) for t in M.gram_schmidt() ]
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Asked: 2022-01-30 13:44:45 +0200

Seen: 36 times

Last updated: Jan 30