Exercise 1.9.3

Since $A$ is itself orthogonal, we have $A=\left[\begin{array}{cc}\hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill -\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \\ \hfill \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \end{array}\right]=U\mathrm{\Sigma }{V}^T=\mathit{AII}$. All its signular values are 1, so the rank-1 approximation is $\left[\begin{array}{c}\hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \\ \hfill \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill 0\hfill \\ \hfill \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill 0\hfill \end{array}\right]$