Exercise 2.3.9

If we set $𝜖=0$, and from problem 8, we see the two orthogonal vectors are approximately $q_1=\frac{1}{\sqrt{3}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$, $q_2=\frac{\sqrt{3}}{\sqrt{2}}\left[\begin{array}{c}\hfill \frac{2}{3}\hfill \\ \hfill -\frac{1}{3}\hfill \\ \hfill -\frac{1}{3}\hfill \end{array}\right]$

and $r_1=\left[\begin{array}{ccc}\hfill \sqrt{3}\hfill & \hfill \sqrt{3}\hfill & \hfill \frac{1}{\sqrt{3}}\hfill \end{array}\right]$ and $r_2=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{\sqrt{3}}\hfill \end{array}\right]$

So we have $A=\mathit{QR}+O(𝜖)=\left[\begin{array}{cc}\hfill \frac{1}{\sqrt{3}}\hfill & \hfill \frac{2}{\sqrt{6}}\hfill \\ \hfill \frac{1}{\sqrt{3}}\hfill & \hfill -\frac{1}{\sqrt{6}}\hfill \\ \hfill \frac{1}{\sqrt{3}}\hfill & \hfill -\frac{1}{\sqrt{6}}\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill \sqrt{3}\hfill & \hfill \sqrt{3}\hfill & \hfill \frac{1}{\sqrt{3}}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\sqrt{2}}{\sqrt{3}}\hfill \end{array}\right]+O(𝜖)=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]+\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 𝜖\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$