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.