Suppose we have a Gram matrix $A$ (symmetric, positive semi-definite).
I cannot recall the exact statement, but there was some theorem like:
For a vector $x$ of unit length $||x||_2 = 1$:
$x^* = \arg \min \limits_x || A - \lambda x x^T ||_F = \arg \max \limits_x \sqrt{ x^T A x } = \arg \max \limits_x || x ||_A $, where:
- $|| A - \lambda x x^T ||_F$ is the Frobenius norm of rank-1 approximation error
- $|| x ||_A = \sqrt{x^T A x}$ is the energy norm of vector $x$ upon matrix $A$ (or, equivalently, same result, formulated in terms of Rayleigh quotient).
Can you suggest the exact formulation of this result and its name?