Linear Algebra: Unitary matrix

[Niles]

Homework Statement

Hi

My teacher told us that if we have a unitary matrix U, then

$$\sum\limits_p {\left| {U_{np} } \right|^2 } = 1$$

Is that really correct? I thought he should be summing over n, not p.

 

[Dick]

Use that if U is unitary, then the hermitian conjugate of U is unitary also to show you can sum over either index.

 

[Niles]

Use that if U is unitary, then the hermitian conjugate of U is unitary also to show you can sum over either index.

Hmm, all I know is that U^-1=U^H. I cannot see how that helps me.

 

[Dick]

Hmm, all I know is that U^-1=U^H. I cannot see how that helps me.

Define V=U^H. Then V also satisfies V^(-1)=V^H. So V is also unitary. The sum over the second index for U is the same as the sum over the first index for V.

 

[Niles]

I see, very smart. Thanks.

Have a nice day.