Let $S$ be the set of all $2 \times 2$ complex matrices $A$ which are hermitian, that is, $A^{*}=A$, where $A^{*}=\bar{A}^{t}$.

(a) Show that $S$ is a real 4-dimensional vector space. Consider the real symmetric bilinear form $b$ on this space defined by

$$b(A, B)=\frac{1}{2}(\operatorname{tr}(A B)-\operatorname{tr}(A) \operatorname{tr}(B)) .$$

Prove that $b(A, A)=-\operatorname{det} A$ and $b(A, I)=-\frac{1}{2} \operatorname{tr}(A)$, where $I$ denotes the identity matrix.

(b) Consider the three matrices

$$A_{1}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right), \quad A_{2}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \quad \text { and } \quad A_{3}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)$$

Prove that the basis $I, A_{1}, A_{2}, A_{3}$ of $S$ diagonalizes $b$. Hence or otherwise find the rank and signature of $b$.

(c) Let $Q$ be the set of all $2 \times 2$ complex matrices $C$ which satisfy $C+C^{*}=\operatorname{tr}(C) I$. Show that $Q$ is a real 4-dimensional vector space. Given $C \in Q$, put

$$\Phi(C)=\frac{1-i}{2} \operatorname{tr}(C) I+i C .$$

Show that $\Phi$ takes values in $S$ and is a linear isomorphism between $Q$ and $S$.

(d) Define a real symmetric bilinear form on $Q$ by setting $c(C, D)=-\frac{1}{2} \operatorname{tr}(C D)$, $C, D \in Q$. Show that $b(\Phi(C), \Phi(D))=c(C, D)$ for all $C, D \in Q$. Find the rank and signature of the symmetric bilinear form $c$ defined on $Q$.