kakarukeys
Aug26-07, 10:56 PM
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.
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.