How to calculate complex dot products

[BOYLANATOR]

let y₁=(1,0,i,0) and y₂=(0,i,1,0)

what is (y₁.y₂)/(y₁.y₁)?

I make it i/(1+i²) = i/0 which seems incorrect.

My notes seem to give the answer as -i/2 and I don't understand how this was calculated.

Any help is appreciated. Thanks

 

[rustynail]

hmm.. I get (i/0) as well.

(1*0 + 0*i + i*1 + 0²) / (1² + 0² + i² + 0² ) =

(0 + 0 + i + 0 ) / (1 + 0 -1 +0) =

(y1*y2)/(y1²)= i/0

Does anyone get a different answer?

 

[BOYLANATOR]

I suspect the answer is to do with the fact that

<z₁,z₂>=<(z₂)^*,z₁> where (z₂)^* is the Hermitian conjugate of z₂, but even if this is the case I'm not sure how to carry out the calculation.

 

[sachav]

It seems to work with y₁ = (i,0,i,0), you probably didn't copy it correctly.

 

[DonAntonio]

let y₁=(1,0,i,0) and y₂=(0,i,1,0)

what is (y₁.y₂)/(y₁.y₁)?

I make it i/(1+i²) = i/0 which seems incorrect.

My notes seem to give the answer as -i/2 and I don't understand how this was calculated.

Any help is appreciated. Thanks

The usual (euclidean) complex dot product in $\mathbb C^n$ is defined as the sum of the

products of the first vector's coordinates times the conjugate of second vector's coordinates.

In your case, $y_1\cdot y_1 = 1\cdot 1 + 0 + i\cdot (-i) +0=2$ , and this is what must appear in the denominator.

DonAntonio

 

[BOYLANATOR]

Thank you DonAntonio
I must have missed this definition. When you apply it to the numerator as well, you get the correct answer -i.
Thanks

 

[DonAntonio]

Thank you DonAntonio
I must have missed this definition. When you apply it to the numerator as well, you get the correct answer -i.
Thanks

No. It is either $y_1\cdot y_2 = i\,\,or\,\,y_2\cdot y_1=-i$. Check this.

DonAntonio

 

[BOYLANATOR]

Yes it was y[2].y[1] = -i . From the Gram-Schmidt Process although I changed the notation slightly.
Thanks