Let $B : \mathbb{R}^4 × \mathbb{R}^4 → \mathbb{R}$ be a non-degenerate bilinear form and let $T : \mathbb{R}^4 → \mathbb{R}^4$ be a linear transformation. We denote by $T^∗$ the linear transformation which satisfies the condition $B(T(v),w)=B(v,T^∗(w))$ for all v, w ∈ V . Prove or disprove the following :
- If $B$ has signature $(3,1)$ and if $T^∗ = T$, then for any $v ∈ V$ such that $T^2(v) = 0$, then $T(v) = 0$.
If $B$ is negative definite then I can prove this by $B(T(v),T(v))=B(v,T^*(T(v)))=B(v,T^2(v))=B(v,0)=0$ which implies $T(v)=0$. But I am not able to prove the case when $B$ has signature $(3,1)$ or $(2,2)$.