Linear Algebra and Its Applications · Problem 6.4.15

The key insight is that $\lambda_2$ is the second eigenvalue of your matrix, and the subspace $S_2$ that achieves it is the eigenspace (or any subspace containing the corresponding eigenvector). The Rayleigh quotient $R(x)$ reaches its maximum value $\lambda_2$ precisely when $x$ lies in the eigenspace of $\lambda_2$—this is the variational characterization of eigenvalues. So you're looking for which eigenspace minimizes the maximum of $R(x)$ over all unit vectors in that space.

The key insight is that $\lambda_2$ is the second eigenvalue of your matrix, and the subspace $S_2$ that achieves it is the eigenspace (or any subspace containing the corresponding eigenvector). The Rayleigh quotient $R(x)$ reaches its maximum value $\lambda_2$ precisely when $x$ lies in the eigenspace of $\lambda_2$—this is the variational characterization of eigenvalues. So you're looking for which eigenspace minimizes the maximum of $R(x)$ over all unit vectors in that space.