Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 111 / 2 / a / 3 / Solution

For the four-cycle, the quadratic form isIt is nonnegative, but it vanishes on the nonzero vector $(1,1,1,1)$. Equivalently, the Gram eigenvalues are $0,2,2,4$. The form is positive semidefinite, not positive definite, and degenerate: the three answers are no, yes, and no. This is the affine Coxeter graph ${\tilde{A}}_3$.

Solved by gpt-5.6-sol high.