Why can we write an arbitrary object  $v_{a \dot{b} }$ our transformations in this basis act on as

$$ v_{a \dot{b} } = v_{\nu} \sigma^{ \nu}_{a \dot{b} } = v^0 \begin{pmatrix}    1&0 \\ 0&1 \end{pmatrix}  + v^1 \begin{pmatrix}    0&1 \\ 1&0 \end{pmatrix}  +v^2 \begin{pmatrix}    0&-i \\ i&0 \end{pmatrix}  + v^3 \begin{pmatrix}   1&0\\0&-1 \end{pmatrix} $$


Formulated differently: How do we know that the vector space for the $(\frac{1}{2},\frac{1}{2})= (\frac{1}{2},0) \otimes (0,\frac{1}{2})$ representation of the Lorentz group is the space of hermitian $2\times2$ matrices? The vector space for the $(\frac{1}{2},0)$ representation is $\mathbb{C}^2$ and I guess the same is true for the $(0,\frac{1}{2})$  representation, but I can't put it together to end up with hermitian matrices. 


EDIT: I found in [this book][1] the following explanation. 

*" Recall that just as any
real matrix can be written as the sum of a symmetric matrix and an antisymmetric
matrix, any complex matrix can be written as the sum of a Hermitian matrix and
an anti-Hermitian matrix. However, the two indices on our matrix $v^{a \dot b}$
transform under representations of $SU(2)$. Notice that in the generators of these copies of $SU(2)$, both sets of generators $N^-$
and $N^+$ are Hermitian (cf. (3.229)). So, we’ll limit our discussion to the case where $v^{a \dot b}$ is a Hermitian $2 \times  2$ matrix."*

**If anyone could help me understand this line of thought my problem would be solved.**

 - First of all, I dont think that $N^-$ and $N^+$ are hermitian, because we have $(N^-)^\dagger=N^+$. 
 - Seconldy, why does this allow us to " limit our discussion to the case where $v^{a \dot b}$ is a Hermitian $2 \times  2$ matrix"?


I understand that our representation here acts on complex $2\times2$ matrices. But I don't understand why we can restrict to hermitian matrices.

  [1]: https://books.google.de/books?id=JcvWry8rjTwC&pg=PA121&lpg=PA121&dq=vector%20representation%20lorentz%20group&source=bl&ots=mdiO9U6yqb&sig=5AVy1UF4J0w-5s0Tw8aqdDWf4kY&hl=de&sa=X&ei=tsPEVOmGNYrfOMPOgMgH&ved=0CDgQ6AEwBDgU#v=onepage&q=vector%20representation%20lorentz%20group&f=false