I've read this answer to a question about Symmetric Rank Update (SR1).
In this approach, we require the update of the Hessian matrix to be of the form $$ \mathbf{B}_{k+1}=\mathbf{B}_{k} + \mathbf{u}\mathbf{u}^T \tag{1} $$ under the constraint $$ \mathbf{B}_{k+1} \mathbf{s}_{k}=\mathbf{y}_{k} \tag{2} $$ Here $\mathbf{y}_{k}=\nabla f(\mathbf{x}_{k}+\mathbf{s}_{k})- \nabla f(\mathbf{x}_{k}) $. Plugging $(1)$ into $(2)$ yields \begin{eqnarray*} \mathbf{B}_{k} \mathbf{s}_{k}+ \mathbf{u}\mathbf{u}^T\mathbf{s}_{k} &=& \mathbf{y}_{k} \\ (\mathbf{u}^T\mathbf{s}_{k}) \mathbf{u} &=& \mathbf{y}_{k}-\mathbf{B}_{k} \mathbf{s}_{k} = \mathbf{e}_k \tag{3} \end{eqnarray*} which basically says that we can write $\mathbf{u}=\alpha\ \mathbf{e}_k$
What I don't understand is why we can write $\textbf{uu}^Ts_k = \textbf{u}^Ts_k\textbf{u}$ and why we can say from that that $\textbf{u}=\alpha \textbf{e}_k$. Could someone add details to this so I can understand?
Thank you.