# Homework Help: Unitary matrices

1. Jun 1, 2010

### Niles

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

Is it correct that when I have a unitary 3x3 matrix U, then

|Un,1|2+|Un,2|2+|Un,3|2=|U1,n|2+|U2,n|2+|U3,n|2,

since UH=U? Here n denotes some integer between 1 and 3.

2. Jun 1, 2010

### MathematicalPhysicist

U=U* is called hermitian matrix not unitary, a unitary matrix satisifies: UU*=I.
If you multiply what do you get?

3. Jun 1, 2010

### Niles

My book says that a unitary matrix satisfies UHU=I, i.e. UH=U-1.

4. Jun 1, 2010

### Cyosis

I don't think so. That is not an example of a unitary matrix that is Hermitian. You just wrote the definition of a unitary matrix in another form.

Definition of a unitary matrix: $$UU^\dagger=I$$. Then we multiply both sides with the inverse of U, which gives us $$(U^{-1}U)U^\dagger=IU^\dagger=U^\dagger=U^{-1}$$.

The definition of a Hermitian matrix is:

$$U=U^\dagger$$

note that it is not the same as the equality you wrote in post #3.

Use the definition of the conjugate transpose $$(A^\dagger)_{ij}=\overline{A}_{ji}$$.