State Sylvester's law of inertia.

Find the rank and signature of the quadratic form $q$ on $\mathbf{R}^{n}$ given by

$$q\left(x_{1}, \ldots, x_{n}\right)=\left(\sum_{i=1}^{n} x_{i}\right)^{2}-\sum_{i=1}^{n} x_{i}^{2}$$