The space $E$ of all complex hermitian $2\times2$ matrices is the space of all matrices
$$
\left(
\begin{array}{cc}
x_0& x_2-ix_3\\
x_2+ix_3&x_1
\end{array}\right)
\quad
x_0,x_1,x_2,x_3\in\R~.
$$
1. Verify that $E$ has real dimension four and $A\mapsto-\det(A)$ is a quadratic form on $E$. Thus there is a unique bi-linear form $g$ on $E\times E$ such that $g(A,A)=-\det(A)$. 2. Show that $g$ is a Lorentz product. 3. The subspace $F\colon=\{A\in E:\tr A=0\}$ is space-like and the identity is a time-like vector.
2. $-\det A=-x_0x_1+x_2^2+x_3^2$, thus the symmetric bi-linear map is given by
$$
(x_0,x_1,x_2,x_3),(y_0,y_1,y_2,y_3)
\mapsto-\tfrac12(x_0y_1+x_1y_0)+x_2y_2+x_3y_3
$$
with Gramian (of the canonical basis):
$$
\left(\begin{array}{cccc}
0&-\frac12&0&0\\
-\frac12&0&0&0\\
0&0&1&0\\
0&0&0&1
\end{array}\right)~.
$$
It's eigenvalues are $-1/2,1/2,1,1$, i.e. $g$ is a Lorentz product. The normalized (with respect to $g$) eigen vectors are given by:
$$
\s_0=\left(
\begin{array}{cc}
1&0\\
0&1
\end{array}\right),
\s_3=\left(
\begin{array}{cc}
1&0\\
0&-1
\end{array}\right),
\s_1=\left(
\begin{array}{cc}
0&1\\
1&0
\end{array}\right),
\s_2=\left(
\begin{array}{cc}
0&-i\\
i&0
\end{array}\right)~.
$$
The matrices $\s_1,\s_2$ and $\s_3$ are called the Pauli spin matrices.
3. For $\tr A=0$ we have $g(A,A)=-\det(A)=x_1^2+x_2^2+x_3^2$, i.e. $g$ is euclidean on $F$.
4. A vector $A\colon=x_0\s_0+x_1\s_3+x_2\s_1+x_3\s_2$ is time-like iff $\det A > 0$ and for $A=1$ we have $\det A=1$.
5. For all $A$ in $F$ we have
$$
g(A,\s_0)=-\tfrac12(x_0+x_1)=-\tfrac12\tr A=0~.
$$