# Unitary Matrix mutually orthonormal vectors

1. Nov 25, 2012

### physics2000

1. The problem statement, all variables and given/known data

Demonstrate that the columns of a unitary matrix form a set of mutually orthonormal vectors.

2. Relevant equations

hint - form the vectors $$u_i = {U_{ji}}$$ and $$u_k={U_{jk}}$$ from the $$i^{th}$$ and $$j^{th}$$ columns of $$U$$ and make use of the relationship $$U^{\dagger}U=I$$

3. The attempt at a solution

I thought my work was following the hint...but not sure...I know I need to end up with an identity matrix from the hint, in which it shows the 3 columns are (1,0,0) , (0,1,0), and (0,0,1), respectively...to show that I have a set of basis vectors in the unitary matrix...

Last edited: Nov 25, 2012
2. Nov 25, 2012

### Dick

It looks like you've got it. The Kronecker delta symbol $\delta_{jk}$ is 1 if j=k and 0 if j≠k. Doesn't that give you your identity matrix?

3. Nov 25, 2012

### physics2000

thanks,

I'm just a little confused because It looks like I only have one vector.

Do I need to turn that vector into a column, and for column 1 let i=k=1 , and for column 2 let i=k=3 and for column 3 let i=k=3, that would give:
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but I'm just not sure if my previous math boiled down to that, I feel like I'm missing something in the proof

4. Nov 25, 2012

### physics2000

thanks,

would this be the next step? I feel like my notation is wrong or I did something wrong in the process