Difference between the A conjugate and A dagger

[Hazzattack]

Hi guys, getting a little confused whilst looking through a paper. I was hoping someone could clear this up for me quickly;

If we have a matrix denoted by the following, which is both real and unitary;

A_mn

then is the following true (due to it being real, the conjugate just produces the same thing);

A_mn^* = A_mn

However, if we dagger A, the indices change like the following;

A_mn⁺ = A_nm

I think I'm getting confused because i expected both the conjugate and the dagger to do the same thing...

I am trying to get to grips with the paper, but some of the summation notation to represent matrices is confusing me. Any good reads someone could recommend to help with this would be great. Otherwise confirmation of what I've previously said would be amazing.

thanks in advance!

 

[Hazzattack]

If the above checks out, could someone please elaborate on the difference between these two things? thanks.

 

[economicsnerd]

Different people adopt different notation, but if I were writing, then for any matrix A, both notations A^*=A^\dagger would refer to the conjugate transpose. That is, a^*_{ij}=a^\dagger_{ij}=\overline{a_{ji}}.

 

[MikeGomez]

i expected both the conjugate and the dagger to do the same thing...

The complex conjugate and dagger do not do the same thing.

The complex conjugate keeps the real part of a number the same, and negates the imaginary part. If the number has only a real part, with no imaginary part, then the complex conjugate is the same as the original.

The transpose of a matrix is the matrix reflected about it’s main diagonal axis (or you could say that you flip the rows with the columns which is the same thing).

The dagger is the complex conjugate of the transpose. In other words, the matrix is conjugated and transposed, not just conjugated.

 

[Robert1986]

Are you sure the matrix is real? Usually if the matrix is real, people say "Orthogonal" instead of "Unitary" and instead of using a star, they use either a T or a dagger (T is to denote transpose.) Does the paper explicitly say that the matrix is real?