Unitary Matrix Property: |Uij|2 = UijU*ji

[Niles]

Hi guys

I have been sitting here for a while thinking of why it is that for a unitary matrix U we have that U_ijU^*_ji = |U_ij|². What property of unitary matrices is it that gives U this property?


Niles.

 

[quasar987]

This is true of any matrix: $$U^*=\overline{U}^T$$. So $$U^*_{ji}=\overline{U_{ij}}$$. And of course, for any complex number z, we have $z\overline{z}=|z|^2$

 

[Niles]

By an asterix I meant complex conjugation, so $$(U^\dagger )_{ij} = (U_{ji})^*$$. Is it still valid then?

 

[quasar987]

It is not valid in this case. Take for example the rotation matrix

cos(t) -sin(t)
sin(t) cos(t)

It is orthogonal, hence unitary. But for any sin(t) different from 0, we have $U_{12}U^*_{21}=-\sin^2(t)\neq |\sin(t)|^2=|U_{12}|^2$.

 

[Niles]

Hmm, I have a problem then. I have a transformation

$$\mathbf{m} = S\mathbf{a},$$

which has the components

$$m_i = \sum_j S_{ij}a_j.$$

Now I want to find the Hermitian conjugate (I denote this by a dagger, and complex conjugation is denoted by an asterix), and we have

$$\mathbf{m}^\dagger = \mathbf{a}^\dagger S^\dagger,$$

which has the components

$$m^\dagger_i &= \sum_j a_j^\dagger (S^\dagger)_{ij} \\ &=\sum_j a_j^\dagger (S^*)_{ji}.$$

My teacher says the last step is wrong, but I cannot see why. Can you help me spot the error?

 

[quasar987]

Your mistake is when you say

$$m^\dagger_i &= \sum_j a_j^\dagger (S^\dagger)_{ij}$$

According to the definition of matrix multiplication, correct is

$$m^\dagger_i &= \sum_j a_j^\dagger (S^\dagger)_{ji}$$

 

[Niles]

Ahh, I see it now. Of course the column has to be fixed, not the row. Thanks.