Ok, ok. I just got it working:

I had to use

convert~(H,rational)

and it worked as I was hoping for: fast and exact. Thanks to @Preben Alsholm for his advice in a previous case.

@Preben Alsholm 

Thank you very much. Your last reply really helped, especially using the ~ for element-wise operation.

If you like, you could also look at my other problem:

http://www.mapleprimes.com/questions/202542-Very-Bad-Eigenvalues-Rounding-Errors

But again, thank you!

@Preben Alsholm 

Thank you for your fast reply. convert(...) did the trick!

However, if you insert a squareroot or Pi at any place, it still works, although those are definitely no rationals but maple works. In contrast, you could define

a := 0.5+0.5*I

and convert(...) would not convert these 0.5 to 1/2 so maple is stuck again... why?