Let $H=\{A \in M_{2\times 2}(\mathbb{C}) \ | \ A = A^\dagger := \overline{A}^t \}$ be the real vector space of all hermitian $2 × 2$ complex matrices.
(a) Show that for all $A \in H$, $\text{det}(A)$ is real-valued.
Define $Q:H \to \mathbb{R}$, by $A \mapsto \text{det}(A)$.
(b) Show that $Q$ is a quadratic form, and calculate the signature of the associated bilinear form $S_Q$.
Let be $U = \{A \in H : \text{Tr}(A) = 0 \} \subseteq H$.
(c) Find $U^\perp$ with respect to $S_Q$.
(d) Calculate the signature of $(S_Q)|_{U \times U} : U \times U \to \mathbb{R}$.
For problem (a), my approach would be that the eigenvalues of hermitian matrices are real-valued. Since the determinant of a matrix $A$ can be represented as the product of the eigenvalues of $A$, the determinant is real-valued, because the factors are all real-valued.
Now I have a problem making a start with task (b), because I don't know how to define $Q$. Could someone give me a hint on how to do this? $$ \left( \begin{array}{rrrr} 0 & 1/2 & 0 & 0 \\ 1/2 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{array} \right) $$