Gram–Schmidt Process for Orthonormalizing Vectors in R^n

[kakarukeys]

http://en.wikipedia.org/wiki/Gram-Schmidt_process

Can Gram–Schmidt process be used to orthonormalize a finite set of linearly independent vectors in a space with any nondegenerate sesquilinear form / symmetric bilinear form not necessarily positive definite?

For example in R^2 define
\langle a, b\rangle = - a_1\times a_1 + a_2\times a_2

From \{v_1, v_2\} to \{e_1, e_2\}, assume v's are not null.
e_1 = \frac{v_1}{|v_1|}
where |v_1| = \sqrt{|\langle v_1, v_1\rangle|}
t_2 = v_2 - \frac{\langle v_1, v_2\rangle}{\langle v_1, v_1\rangle}v_1
e_2 = \frac{t_2}{|t_2|}

It looks like it can be generalized to R^n without any problem.

 

[radou]

In general, as long as you have a valid inner product, it works.

 

[kakarukeys]

Yes, I'm asking if we drop the assumption of positive definiteness of inner product, will it work?

 

[mathwonk]

well to get unit length vectors you divide by the length.