Assume now that we have a symmetric positive-definite matrix $\hat{A}$ of size $n\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector

$$\begin{equation*} \hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. \end{equation*}$$

We assume that $\hat{p}_{i}$ is a sequence of $n$ mutually conjugate directions. Then the $\hat{p}_{i}$ form a basis of $R^n$ and we can expand the solution $ \hat{A}\hat{x} = \hat{b}$ in this basis, namely

$$\begin{equation*} \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i. \end{equation*}$$