Fredy Yip

Tight bounds for positive discrepancy via eigenvalues

Given an $n\times n$ symmetric matrix $M$ with largest eigenvalue $\lambda_1\geq 0$, it is easy to show that the solution of the optimisation problem $\max_{v\in [-1,1]^n}v^TMv$ is at most $\lambda_1 n$. We prove the following converse: if every $n'\times n'$ principal submatrix of $M$ has maximal eigenvalue at least $...