# Expanding subspace theorem

Let $LaTeX: S_k$ be the subspace spanned by vectors $LaTeX: d^k$, which are Q-conjugate. Let $LaTeX: x^0$ be any initial point in $LaTeX: \mathbb{R}^n$, and let $LaTeX: x^k$ be generated by these conjugate directions with optimal line search:
$LaTeX: \displaystyle x^{k+1} = x^k + s_k d^k \;\; \mbox{ and } \;\;
s_k = \frac{-(Qx^k +c)}{(d^k)^T Q d^k}$
Then, $LaTeX: x^k$ minimizes $LaTeX: x^{T} Q x + c^{T} x$ on the affine set, $LaTeX: x^0 + S_k$.