Unitary Matrices: Properties & Homework Solutions

[Niles]

Homework Statement

Hi

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

|U_n,1|²+|U_n,2|²+|U_n,3|²=|U_1,n|²+|U_2,n|²+|U_3,n|²,

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

 

[MathematicalPhysicist]

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

 

[Niles]

My book says that a unitary matrix satisfies U^HU=I, i.e. U^H=U^-1.

 

[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 $$(A^\dagger)_{ij}=\overline{A}_{ji}$$.