Given an orthogonal projection method (when $K = L$) and a symmetric positive definite (SPD) matrix $A$, show that $\tilde{x}$ minimizes $(A(x−x^*),(x−x^*))≡ \|x - x^* \| ^{2} _A = ( A ( x − x^* ) , ( x − x^* ) ) $ over all vectors $x$ in the affine space $x_0 +$ where $x^*$ is the exact solution (i.e., $Ax^* = b$).
Here is what I have:
The Galerkin condition for this projection framework gives us $$ b - A \tilde{x} \perp K $$ Substitute in $b = Ax^*$: $$ Ax^* - A \tilde{x} \perp K $$ $$ A(x^* - \tilde{x}) \perp K $$
So $(x^* - \tilde{x})$ is A-orthogonal to $K$. But how does that imply that $\tilde{x} \in x_0 + K$ is the minimizer for the error? In other words, why must $(x^* - \tilde{x})$ be A-orthogonal to $K$ to minimize $ \|x - x^* \| _A$?