Off-diagonal all-ones quadratic form ($q(x)=({\sum }_i{x}_i{)}^2-{\sum }_i{x}_i^2$)

Its symmetric matrix is $11^T-I$. The line spanned by $1$ has eigenvalue $n-1$ and its orthogonal complement has eigenvalue $-1$. Thus for $n>1$ it is nondegenerate with inertia $(1,n-1,0)$; for $n=1$ it vanishes. This gives a useful explicit example for Sylvester's law of inertia.