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 \( \mathbf{e^{T}} = \{1,0,0,\dots 0\} \).