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:
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.
2) Verify that the output of sage is indeed correct.
Gram-Schmidt orthogonalization is an algorithm that takes in a basis as an input and returns a basis where all vectors are orthogonal, i.e. at right angles. This new basis is defined as
where is the Gram-Schmidt coefficients.
Let be the matrix where the th row is given by andbe the matrix where the th row is given by , then the Gram-Schmidt orthogonalization gives us where and is the Gram-Schmidt coefficient. As an example, consider the basis of a subspace of :
This outputs two matrices, and :
One can quickly verify that and that the rows of are orthogonal to each other.
1) Show that the basis is orthogonal.
3) Show that and is a diagonal matrix whose entries are . Conclude that .
4*) Given the Iwasawa decomposition where is a lower diagonal matrix with on its diagonal, is a diagonal matrix and an orthogonal matrix, meaning , show that and . Furthermore, prove that such a decomposition is unique.