How to make the conjugate transpose

Question

I have a problem to make the conjugate and transpose the matrix.May anyone help me please.

MatrixForm[{{E^(I*β1 + I*β3) Cos[β2], 
   E^(I β1 - 
       I*β3) Sin[β2]}, {(-E^((-I) β1 + 
         I*β3)) Sin[β2], 
   E^((-I) β1 - I*β3)*Cos[β2]}}]

and I get the output

{{E^((-I)*Conjugate[β1] - I*Conjugate[β3])*
   Conjugate[Cos[β2]], 
  -(E^(I*Conjugate[β1] - I*Conjugate[β3])*
    Conjugate[Sin[β2]])}, 
 {E^((-I)*Conjugate[β1] + I*Conjugate[β3])*
   Conjugate[Sin[β2]], 
  E^(I*Conjugate[β1] + I*Conjugate[β3])*
   Conjugate[Cos[β2]]}}

my problem here, I just want to conjugate the imaginary part but it conjugate all include the angle Cos and Sin (real). How can I solve this.

Thank you

Answer 1

Use Simplify with Assuming:

m = {{E^(I*β1 + I*β3) Cos[β2], 
    E^(I β1 - I*β3) Sin[β2]}, {(-E^((-I) β1 + I*β3)) Sin[β2], 
    E^((-I) β1 - I*β3)*Cos[β2]}};

MatrixForm[
 Assuming[{β1, β2, β3} ∈ Reals, 
  Simplify@ConjugateTranspose[m]]]

$$\left( \begin{array}{cc} e^{i \text{$\beta $1}+i \text{$\beta $3}} \cos (\text{$\beta $2}) & e^{i \text{$\beta $1}-i \text{$\beta $3}} \sin (\text{$\beta $2}) \\ -e^{i \text{$\beta $3}-i \text{$\beta $1}} \sin (\text{$\beta $2}) & e^{-i \text{$\beta $1}-i \text{$\beta $3}} \cos (\text{$\beta $2}) \\ \end{array} \right)$$