This problem has me completely stumped.
Given:
$A$ is a symmetric $n \times n$ matrix
$r = b - Ax$ with $r, b, x \in \mathbb{R}^n$ and $x$ is nonzero.
Show how to compute a symmetric $E \in \mathbb{R}^{n\times n}$ with minimal Frobenius norm so that $\left( A+E \right) x = b$.
Hint: Use the $QR$ factorization of $[x, r]$ and note that $Ex=r \Rightarrow \left( Q^T E Q \right) \left( Q^T x \right) = Q^T r $
Hint: First show that $E$ has minimal Frobenius norm iff $Range(E) = Span\left\{x, r\right\}$
I've filled three pages with scratch work and gotten nowhere. Obviously setting $E=\frac{r x^T}{\Vert x \Vert_{2}^{2}} $ satisfies $\left( A+E \right) x = b $ but the norm is not minimized. Any hints would be appreciated