Complex number sum that should be easy

[trap101]

Hey,

So I have a sum of complex numbers that really should be easy, but I'm not getting the right solution. It is with respct to using the Gram Schmidt process

U₁ = (i, -1, i) U₂ = (1,1,0)

So I perform the Gram Schmidt with U₁ being my initial vector selection and I get:

V₂ = (1,1,0) - [(-1 + i)/3 ] (i, -1 , i)

and here is where the issue arises. Now I did regular subtraction w.r.t complex numbers and got:

[ (4 + i)/ 3, (2+i)/3 , (1+i)/3]

but the solution is:

[(2+i)/3 , (2-i)/3, (-1+i)/3 ]

my vector is not orthogonal, so my question is what am I missing with respect to this?

thanks

 

[Dick]

Hey,

So I have a sum of complex numbers that really should be easy, but I'm not getting the right solution. It is with respct to using the Gram Schmidt process

U₁ = (i, -1, i) U₂ = (1,1,0)

So I perform the Gram Schmidt with U₁ being my initial vector selection and I get:

V₂ = (1,1,0) - [(-1 + i)/3 ] (i, -1 , i)

and here is where the issue arises. Now I did regular subtraction w.r.t complex numbers and got:

[ (4 + i)/ 3, (2+i)/3 , (1+i)/3]

but the solution is:

[(2+i)/3 , (2-i)/3, (-1+i)/3 ]

my vector is not orthogonal, so my question is what am I missing with respect to this?

thanks

You are probably missing that dot product of complex vectors involves taking a complex conjugate of one of the vectors. And <u1,u2> is not necessarily equal to <u2,u1>. Do it again and be careful.

 

[Mark44]

Hey,

So I have a sum of complex numbers that really should be easy, but I'm not getting the right solution. It is with respct to using the Gram Schmidt process

U₁ = (i, -1, i) U₂ = (1,1,0)

So I perform the Gram Schmidt with U₁ being my initial vector selection and I get:

V₂ = (1,1,0) - [(-1 + i)/3 ] (i, -1 , i)

and here is where the issue arises. Now I did regular subtraction w.r.t complex numbers and got:

[ (4 + i)/ 3, (2+i)/3 , (1+i)/3]

but the solution is:

[(2+i)/3 , (2-i)/3, (-1+i)/3 ]

my vector is not orthogonal, so my question is what am I missing with respect to this?

thanks

I checked the solution vector by dotting u₁ with (their) v₂, and got a value of zero. Are you forgetting that if u and v are complex vectors, then $u \cdot v = \sum u_i * \overline{v_i}$? IOW, you need to use the complex conjugate?

Show us how you calculated v₂.

 

[HallsofIvy]

Also, you are dividing by "3" when you should be dividing by \sqrt{3}.

 

[trap101]

This is how I got my V₂


V₂ = U₂ - {[<(i, -1 , i) , (1,1,0)>]/ [<(i, -1 , i), (i, -1 , i)>]} (i, -1, i)


ok, well I figured it out, and well I'm having a big conundrum here. Essentially it came down to the order in which I performed the hermitian inner product. But I had dubbed
U₁ = (i, -1 , i) and when I used the GS formula I did my inner product as:

<V₁, U₂ >. Doing it that way I got the wrong result, but doing it <U₂, V₁> I got the right result. Which brings up the question, of which is the way it is supposed to be and is that way fixed? Because when doing the inner product w.r.t just the reals this problem doesn't arise.

Thanks for the help.