Ideal and worst-case GMRES coincide for a Jordan block

For every $n\ge2$, every $\lambda\in\mathbb C\setminus\{0\}$ and every $1\le k<n$, the Jordan block $J_n(\lambda)=\lambda I+N$ satisfies $\max_{\|v\|_2=1}\min_{p\in\mathcal P_k}\|p(J_n(\lambda))v\|_2=\min_{p\in\mathcal P_k}\|p(J_n(\lambda))\|_2$, where $\mathcal P_k$ is the set of complex polynomials of degree at most $k$ with $p(0)=1$: worst-case and ideal GMRES coincide. The argument proves a stronger minimax equality for finite affine families of lower triangular Toeplitz matrices. Finite Carathéodory–Fejér interpolation describes the maximal right singular subspace of a triangular Toeplitz matrix through a finite Blaschke product, and scalar polynomial spectral factorization replaces a convex combination of maximal singular vectors by a single vector preserving every complex GMRES orthogonality equation. No simplicity, divisibility or eigenvalue-regime restriction is needed.