First step of the improvement

Assuming n>1, we can add some amount of the previous step ${\bf s}_{n-1}$ to the chosen direction ${\bf c}_n$ to produce a new search direction ${\bf s}_n^{(n-1)}$, as follows:  

$${\bf s}_n^{(n-1)} = {\bf c}_n + \beta_n^{(n-1)}\,{\bf s}_{n-1}\;,$$ (10)

where $\beta_n^{(n-1)}$ is an adjustable scalar coefficient. According to to the fundamental orthogonality principle (7),  

$$\left({\bf r}_{n-1},\,{\bf A\,s}_{n-1}\right) = 0\;.$$ (11)

As follows from equation (11), the numerator on the right-hand side of equation (9) is not affected by the new choice of the search direction:  

$$\left({\bf r}_{n-1},\,{\bf A\,s}_n^{(n-1)}\right)^2 = \left[... ...ight)\right]^2 = \left({\bf r}_{n-1},\,{\bf A\,c}_n\right)^2\;.$$ (12)

However, we can use transformation (10) to decrease the denominator in (9), thus further decreasing the residual ${\bf r}_n$. We achieve the minimization of the denominator  

$$\Vert{\bf A\,s}_n^{(n-1)}\Vert^2 = \Vert{\bf A\,c}_n\Vert^2 ... ... + \left(\beta_n^{(n-1)}\right)^2\,\Vert{\bf A\,s}_{n-1}\Vert^2$$ (13)

by choosing the coefficient $\beta_n^{(n-1)}$ to be  

$$\beta_n^{(n-1)} = - {{\left({\bf A\,c}_n,\,{\bf A\,s}_{n-1}\right)} \over {\Vert{\bf A\,s}_{n-1}\Vert^2}}\;.$$ (14)

Note the analogy between (14) and (6). Analogously to (7), equation (14) is equivalent to the orthogonality condition  

$$\left({\bf A\,s}_n^{(n-1)},\,{\bf A\,s}_{n-1}\right) = 0\;.$$ (15)

Analogously to (8), applying formula (14) is also equivalent to defining the minimized denominator as  

$$\Vert{\bf A\,c}_n^{(n-1)}\Vert^2 = \Vert{\bf A\,c}_n\Vert^2 ... ... A\,s}_{n-1}\right)^2} \over {\Vert{\bf A\,s}_{n-1}\Vert^2}}\;.$$ (16)