Question on the Gram-Schmidt orthonormalization process?

Question

Say our general eigenvector for the eigenpolynomial (x-2)(x-1)^4 is: (a+b, a-2b, ....)

We know
v1 = u1
and
v1 . v2 = 0

so I set up the equation
v1 = (a + b)
v2 = (a - 2b) - ((a-2b) (dotted into) (a+b) )/ ((a-2b) (dotted into) (a-2b) ) * (a-2b)

The v2 I got was (6ab^2 - 12b^3) / (a^2 + 4b^2)

multiply by the denominator i got
v2 = 6ab^2 - 12b^3

However, my v1 . v2 doesnt equal to zero.

Is this the right way to do it? or am i doing something wrong?

Answer 1

Yeah, you're doing something wrong. The correct formula for orthogonalization is v₂ = u₂ - (u₂·v₁/v₁·v₁)v₁. What you have is v₂ = u₂ - (u₂·v₁/u₂·u₂)u₂.