steepest descent quadratic single convergence

Question

Is it possible for a quadratic positive definite function f(x) given by $x^TQx + b^Tx + c$ for f() in $R^n \to R$ to converge in 1 iteration to its minimum $x^*$ using the steepest descent algorithm with exact line search if the condition (Q) is not equal to 1 and if you start on one of the major axes of the ellipsoid contours?

Answer 1

This is a simple proof upon further inspection. I solve in limited parts below:

1. let $x_0 = x^* + v$ where $x^*$ is the global minimum and v is an eigenvector on Q.

2. Prove that $\nabla f(x_0) = \lambda v$ easily shown via taylor expansion of the gradient function f'(x)

3. Prove that exact line search $a_0$ = $1/\lambda$ easily shown using the definition of exact line search with quadratic PD

4. Thus it is shown that $x_1 = x_0 + ap$ where p is the $-\nabla f(x)$ using SDM will yield x* in 1 iteration; $x^* + v - \lambda v * 1/\lambda = x^*$