We specialize to a symmetric $3\times 3$ matrix $\mathbf{A}$. We start the process as follows (assuming that $a_{23}=a_{32}$ is the largest non-diagonal) with $c=\cos{\theta}$ and $s=\sin{\theta}$ $$\mathbf{B} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c & -s \\ 0 & s & c \end{array} \right)\left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right) \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c & s \\ 0 & -s & c \end{array} \right).$$ We will choose the angle $\theta$ in order to have $a_{23}=a_{32}=0$. We get (symmetric matrix) $$\mathbf{B} =\left( \begin{array}{ccc} a_{11} & a_{12}c -a_{13}s& a_{12}s+a_{13}c \\ a_{12}c -a_{13}s & a_{22}c^2+a_{33}s^2 -2a_{23}sc& (a_{22}-a_{33})sc +a_{23}(c^2-s^2) \\ a_{12}s+a_{13}c & (a_{22}-a_{33})sc +a_{23}(c^2-s^2) & a_{22}s^2+a_{33}c^2 +2a_{23}sc \end{array} \right).$$ Note that $a_{11}$ is unchanged! As it should.