Let $v\in \mathbb{C}^n$ be an $n$ dimensional complex vector. Define the non-standard bilinear form $\left< u,v \right> = u^T v$ (the usual inner product except without the conjugation). What are the properties of the self orthogonal vectors, those such that $\left < v,v \right>=0$? What are the properties of this space of vectors?
For example, if we say that $v^Tv = 0$ has the null property, then so does $\lambda v$ for any complex $\lambda$, and so does $v^*$ (the conjugated vector). Clearly, the set of null vectors is not a vector space since generally $u+v$ is not null if $u$ and $v$ are null.
This may be well known, but I just have no idea what to Google for. Any pointers to existing literature are appreciated.
Best Answer
The most natural way to view your set of vectors is in the setting of projective geometry; the vector space $K^n$ (where $K$ is now an arbitrary field, so $K = \mathbb{C}$ for you) can be seen as a projective space $\mathbb{P}^{n-1}(K)$ of dimension $n-1$. (The "points" of the projective space $\mathbb{P}^{n-1}(K)$ correspond to the vector lines $\{ \lambda v \mid \lambda \in K \}$ of $K^n$).
In your example, the set of isotropic vectors for the bilinear form $\langle u, v \rangle = u^T v$ correspond to a quadric in $\mathbb{P}^{n-1}(K)$. See, for instance, http://en.wikipedia.org/wiki/Quadric_(projective_geometry). Note that over an algebraically closed field such as $\mathbb{C}$, there is essentially only one non-degenerate bilinear form, so also only one quadric.