Note that $\mathbf{u}\mathbf{u}^T$ is an outer product giving a matrix of dimension ($(n-1) \times (n-1)$). Each matrix element of $\mathbf{P}$ then reads $$P_{ij}=\delta_{ij}-2u_iu_j,$$ where $i$ and $j$ range from $1$ to $n-1$. Applying the transformation $\mathbf{S}_1$ results in $$\mathbf{S}_1^T\mathbf{A}\mathbf{S}_1 = \left( \begin{array}{cc} a_{11} & (\mathbf{Pv})^T \\ \mathbf{Pv}& \mathbf{A}' \end{array} \right) ,$$ where $\mathbf{v^{T}} = \{a_{21}, a_{31},\cdots, a_{n1}\}$ and $\mathbf{P}$s must satisfy ($\mathbf{Pv})^{T} = \{k, 0, 0,\cdots \}$. Then $$\begin{equation} \mathbf{Pv} = \mathbf{v} -2\mathbf{u}( \mathbf{u}^T\mathbf{v})= k \mathbf{e}, \tag{6} \end{equation}$$ with $\mathbf{e^{T}} = \{1,0,0,\dots 0\}$.