I want to verify that what I am doing is correct.
Let $V = \{A \in M_2(\mathbb{R}) : \text{Tr}(A) = 0\}$ and $\langle - , - \rangle$ be the symmetric bilinear form $\langle A, B \rangle = \text{Tr}(AB)$. The set $\mathbb{B} = \{e_1, e_2, e_3\}$ where $e_1 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, e_2 = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, e_3 = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$ is a basis for $V$ that is orthogonal with respect to $\langle - , - \rangle$.
Now I easily compute $\langle e_1 , e_1 \rangle = 2, \langle e_2 , e_2 \rangle = 0, \langle e_3 , e_3 \rangle = 0$. Therefore the signature of $\langle - , - \rangle$ is $(1,0,2)$.
I would like to know if this is right and/or if there are any other approaches to take.