Suppose $A$ is a positive definite symmetric matrix, and K is a subspace. Let $y$ be any minimizer of $x^TAx$, where $\ x\in x_0 + K$.
Then $Ay\perp K$.
It is clear true when $A$ is identity, but I have no idea how to prove the general case. I was also wondering if the positive definite and symmetric conditions of $A$ are both necessary.
Thanks.
PS: The question can be also stated as $$ \text{min}\ y^TAy$$ $$ \text{subject to}\ By = c$$ where $K = \text{ker}\ B$ and $Bx_0 = c$.
One of the KKT conditions is $$2 A y + B^\top \lambda = 0$$ for some vector $\lambda$. Rearranging shows that $Ay$ is in the column space of $B^\top$, which is the same as the row space of $B$, which is orthogonal to $\ker B = K$.