Gram-Schmidt Orthogonalization

Overview

One can immediately check that this new basis is orthogonal, meaning

Instead of doing the Gram-Schmidt orthogonalization by hand, we can get sage to do it for us:

B = Matrix([
[-1, -2, 3, 1],
[-6, -4, 5, 1],
[5, 5, 1, -3]])

B.gram_schmidt()
(
[-1 -2  3  1]  [ 1  0  0]
[-4  0 -1 -1]  [ 2  1  0]
[ 0  3  3 -3], [-1 -1  1]
)

A useful result is that

Intuitively, this tells us that the more orthogonal a set of basis for a lattice is, the shorter it is as the volume must be constant.

Exercises

2) Verify that the output of sage is indeed correct.

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