Exercise 1.9.5

We have $\mathit{QA}=\left[\begin{array}{cc}\hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill -\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \\ \hfill \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \end{array}\right]\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill a\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃-c\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill b\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃-d\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \\ \hfill a\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃+c\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill b\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃+d\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \end{array}\right]$

Take $𝜃=\frac{\pi }{4}$, we have $\mathit{QA}=\left[\begin{array}{cc}\hfill a-c\hfill & \hfill b-d\hfill \\ \hfill a+c\hfill & \hfill b+d\hfill \end{array}\right]$, so

$\left|\right|\mathit{QA}|{\text{|}}_{\infty }=\mathit{max}(|a-c|+|b-d|,|a+c|+|b+d|)$ which doesn’t equal to $\left|\right|A|{\text{|}}_{\infty }$