Optimizing Rayleigh quotient with symmetric constraints and their applications to perturbations of the structured polynomial eigenvalue problem

For a Hermitian matrix $H\in {\mathbb{C}}^{n,n}$ and symmetric matrices $S_0, S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the supremum of $\left{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus {0},,v^TS_iv=0~$\text{for}$~i=0,\ldots,k\right}$. For this, we derive an estimation in the form of minimizing the second largest eigenvalue of a parameter depending Hermitian matrix, which is exact when the eigenvalue at the optimal is simple. The results are then applied to compute the eigenvalue backward errors of higher degree matrix polynomials with T-palindromic, T-antipalindromic, T-even, T-odd, and skew-symmetric structures. The results are illustrated by numerical experiments.