I Generic Form of Hermitian Matrix

[thatboi]

I am reading the following thesis: https://www.kip.uni-heidelberg.de/Veroeffentlichungen/download/6387/pdf-6387.pdf
Specifically, I am confused about equation (2.5), where they give the generic form of the matrix $\mathcal{M}$ due to the Hermiticity of $\mathcal{H}$ and the commutation relation (2.4). I am confused about why the bottom right element is $\bar{A}$. I'm sure this is related to the commutation relation but I'm confused as to how they enter into the picture. $\mathcal{H}^{\dagger} = (\mathcal{M}a)^{\dagger}(a^{\dagger})^{\dagger} = a^{\dagger}\mathcal{M}^{\dagger}a$ so where do the commutation relations come from or what step did I skip?

 

[Hill]

Yes, something is missing, because I can take $N=1$ and $M=\begin{pmatrix} 0&1\\1&2 \end{pmatrix}$, which make $H=a^\dagger a^\dagger + aa+2aa^\dagger$, which is hermitian.