The one-column spectral minimax identity

For $n\ge3$ and a positive spectrum $\lambda$, let $x_1(\lambda)$ be the largest, over orthogonal $V$, of the least trace error of a one-column Nyström approximation of $K_V=V^{\mathsf T}\operatorname{diag}(\lambda)V$, and let $y_1(\lambda)=2e_2(\lambda)/e_1(\lambda)$. The note proves $x_1(\lambda)=y_1(\lambda)=\sum_j\lambda_j-\sum_j\lambda_j^2/\sum_j\lambda_j$ for every positive spectrum, so $R_{n,1}=1$ for every $n\ge3$. The upper bound averages the column errors with weights $K_{ii}/\operatorname{tr}K$. The matching construction uses the lemma that every trace-zero real symmetric matrix has an orthonormal basis in which its diagonal vanishes; applied to $\Lambda^2-(\sum_j\lambda_j^2/\sum_j\lambda_j)\Lambda$ with $\Lambda=\operatorname{diag}(\lambda)$, it gives an orthogonal basis in which every single column leaves the same trace error. The cases $2\le k\le n-2$ are not treated.