Gram-Schmidt Process: Example 3 in PDF

[morsel]

Homework Statement

Example 3 in this pdf: http://karin.fq.uh.cu/qct/Tema_04/0...es propios de un sistema/Diagonalization.pdf"

Homework Equations

Gram-schmidt process:
v2 perp = v2 - (u1*v2)u1

The Attempt at a Solution

I don't understand how they got the second orthonormal basis vector. Using the equation above, with u1=1/sqrt(2)[-1 1 0] and v2=[-1 0 1] (these vectors are vertical), I don't get what they got as the second basis vector. Am I doing the Gram-Schmidt process the wrong way?

 

[vela]

Any non-zero multiple of an eigenvector is also an eigenvector. The results in the examples have been normalized. Did you normalize yours?

 

[morsel]

The examples state that the normalization has been done through Gram-Schmidt but I don't get the same results when I try to normalize them with Gram-Schmidt. Is there another way of normalizing?

 

[vela]

Gram-Schmidt doesn't normalize; it orthogonalizes.

 

[morsel]

Okay, so if I normalize [-1 1 0], I get 1/sqrt(2)[-1 1 0] which is what the examples indicate. However, if I normalize [-1 0 1], I get 1/sqrt(2)[-1 0 1], which doesn't match the example. I don't get how they got 1/sqrt(6)[-1 -1 2] from [-1 0 1].

 

[vela]

Try normalizing [-1 1 2].

 

[morsel]

Where did you get [-1 1 2]? Why can't 1/sqrt(2)[-1 0 1] be a vector in the orthonormal basis?

 

[vela]

Let's back up a second. Did you calculate the second (unnormalized) vector using the Gram-Schmidt process yet? It's this vector that you want to normalize, not the given vector [-1 0 1].

Also, I meant [-1 -1 2] earlier, not [-1 1 2]. I accidentally dropped a sign. You mentioned [-1 -1 2] in your earlier post.

 

[morsel]

Using the Gram-Schmidt process, I get [-1/2 -1/2 1] as the unnormalized second vector. So am I allowed to multiply this by 2 and normalize that to get the second vector in the orthonormal basis?

 

[vela]

Yup, though you don't need to multiply by 2 first. You can normalize it as is.

 

[morsel]

Okay. Thank you so much for working through this with me!