Coefficientwise weighted Frobenius backward error for a polynomial eigenpair

Let $P(\lambda )={\sum }_{j=0}^m{A}_j{\lambda }^j$ be regular of grade $m$, let $[\alpha :\beta ]$ be a projective parameter, let $x\in {\mathbb{C}}^n\setminus \{0\}$, and choose positive weights ${\omega }_0,\dots ,{\omega }_m$. Write $r:=P_h(\alpha ,\beta )x$. The coefficientwise weighted Frobenius backward error is the infimum ${\eta }_F(P,[\alpha :\beta ],x;\omega ):=\inf \{{\left({\sum }_{j=0}^m\frac{\mathrm{\parallel }\mathrm{\Delta }{A}_j{\mathrm{\parallel }}_F^2}{{\omega }_j^2}\right)}^{1\mathrm{/}2}:{\sum }_{j=0}^m(A_j+\mathrm{\Delta }{A}_j){\alpha }^j{\beta }^{m-j}x=0,\mathrm{\Delta }{A}_j\in {\mathbb{C}}^{n\times n}\}\mathrm{.}$ The perturbations of distinct coefficients are unrestricted and measured in the displayed weighted product norm; no symmetry or other coefficient structure is imposed. Replacing $(\alpha ,\beta )$ by $(c\alpha ,c\beta )$ for $c\mathrm{\ne }0$ multiplies the constraint by $c^m$ and leaves its zero set unchanged, so the infimum depends only on $[\alpha :\beta ]$. The infimum is defined for finite and infinite projective parameters. The next theorem proves that it is attained and gives its exact value.