In the CG method we define so-called conjugate directions and two vectors \( \hat{s} \) and \( \hat{t} \) are said to be conjugate if

$$ \begin{equation*} \hat{s}^T\hat{A}\hat{t}= 0. \end{equation*} $$

The philosophy of the CG method is to perform searches in various conjugate directions of our vectors \( \hat{x}_i \) obeying the above criterion, namely

$$ \begin{equation*} \hat{x}_i^T\hat{A}\hat{x}_j= 0. \end{equation*} $$

Two vectors are conjugate if they are orthogonal with respect to this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).