Complex matrix to Block Matrix

[Gaso]

How can I redefine my Complex matrix to a Block matrix, similar as matrix representation of complex number.
I need a Real an Imaginary part as real numbers, to find eigenvalues and eigenvectors with my solver, which works only with Real matrices.

My matrix element is:
$$M_{mn}=\int^{tk}_{0} -i Exp[i E_{mn} t] V_{mn}(t) dt$$
The integrals are easy to write, but are complex.
I want to write matrix M as new matrix of dimension 2N to find eigenvectors and eigenvalues of that matrix, which are also complex.

 

[fresh_42]

We basically have the situation $M=A+iB$. In order to diagonalize $M$ or to use some normal form, we need to know something about the components $A$ and $B$. If we find a normalization for $A$, it could well ruin the structure of $B$ and vice versa. So in any case, the question isn't answerable in this generality. Moreover, complex calculations are usually better than real ones, since we have more eigenvalues available.