A Unitary Matrix and Hermitian Matrix

[Demon117]

Its true that one can say a unitary matrix takes the form

U=e^{iH}

where H is a Hermitian operator. Thats great, and it makes sense, but how can you compute the matrix form of H if you know the form of the unitary matrix U. For example, suppose you wanted to find H given that the unitary matrix is one of the familiar rotation matrices (2 x 2) for simplicity. Let's say

U=\left(\begin{array}{cc} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{array}\right)

What would the procedure be in finding the matrix form of H? I suppose you could start by finding the eigensystem of the unitary matrix. Then, upon normalizing the eigenbasis of U, somehow you could find the matrix representation of H. Any pointers or suggestions would be great.

 

[micromass]

Can you do it for diagonal matrices??

To extend it to nondiagonal matrices, notice that if D=e^{iH}, then

ADA^{-1}=e^{iAHA^{-1}}

 

[Demon117]

So after some fiddling I find that the appropriate Hermitian matrix takes the form

H=\left(\begin{array}{cc} 0 & i\theta \\ -i\theta & 0 \end{array}\right)

If this is indeed correct then I think I have what I need.