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.