rank of quadrics

Question

Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first I see that this quadric can be written as $s^T A s$ with $s=(x,y,z,w)$ and $A$ having entries $a_{14}=-a_{23}=-a_{32}=a_{41}=1/2$ and all other zero. From this I can conclude, since $\mathrm{rank} \ A=4$, that the first quadric has rank $4$ and similarily that the second one has rank $6$. So is this correct, both the result and the approach? Talking from experience, what would you say is the shortest way to determine the rank of a quadric? (I should note that I am mostly interested in a computational, i.e. linear algebra approach.)

Answer 1

a) The rank of any smooth quadric in $\mathbb P^n(\mathbb C)$ is $n+1$.

b) If the quadric $Q$ is not smooth, an elementary algorithm due to Gauss (who else?) linearly transforms its equation to $x_0^2+\ldots +x_r^2=0\quad (r\lt n)$ and its rank is then $r+1$.
In that case the quadric $Q$ is a generalized cone:
$\bullet$ Its base is the smooth quadric $Q_0\subset \mathbb P^r(\mathbb C)=\{x_{r+1}=\ldots=x_n=0\}\subset \mathbb P^n(\mathbb C)$ given by the equations $x_{r+1}=\ldots=x_n=x_0^2+\ldots +x_r^2=0$
$\bullet \bullet$ Its generalized vertex is the linear subspace $ \Sigma_{n-r-1}=\{x_{0}=\ldots=x_r=0\}\subset \mathbb P^n(\mathbb C)$.
That linear subspace $ \Sigma_{n-r-1}\cong \mathbb P^{n-r-1}(\mathbb C) $ is exactly the set of singular points of the quadric $Q$ and it is the union of all the lines joining some point of $Q_0$ to some point of $ \Sigma_{n-r-1}$.