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 $\hat{s}$.