Unitary matrices

[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 UijU*ji = |Uij|2. What property of unitary matrices is it that gives U this property?

Best,
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.