# Gram–Schmidt process

1. Aug 26, 2007

### 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.

Last edited: Aug 26, 2007
2. Aug 27, 2007

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

3. Aug 27, 2007

### kakarukeys

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

4. Aug 27, 2007

### mathwonk

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