(a) Choose \sin {\theta} and \cos {\theta} to triangularize A, and find R:
Givens rotation \quad {Q}_{21} A =\begin{bmatrix} \cos { \theta } & -\sin { \theta } \\ \sin {\theta } & \cos { \theta } \end{bmatrix}\begin{bmatrix} 1 & -1 \\ 3 & 5 \end{bmatrix}=\begin{bmatrix} * & * \\ 0 & * \end{bmatrix}=R.
(b) Choose \sin {\theta} and \cos {\theta} to make QA{Q}^{-1} triangular. What are the eigenvalues?
(a) \cos {\theta} = 1/\sqrt{10},
\sin {\theta} = -3/\sqrt{10},
R=\frac{1}{\sqrt{10}}\begin{bmatrix} 1 & 3 \\ -3 & 1 \end{bmatrix}\begin{bmatrix} 1 & -1 \\ 3 & 5 \end{bmatrix}=\frac{1}{\sqrt{10}}\begin{bmatrix} 10 & 14 \\ 0 & 8 \end{bmatrix}.
(b) A has eigenvalues 4 and 2. Put one of the unit eigenvectors in row 1 of Q:
either Q = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} and QA{Q}^{-1} =\begin{bmatrix} 2 & -4 \\ 0 & 4 \end{bmatrix}
or Q =\frac{1}{\sqrt{10}}\begin{bmatrix} 1 & -3 \\ 3 & 1 \end{bmatrix} and QA{Q}^{-1} =\begin{bmatrix} 4 & -4 \\ 0 & 2 \end{bmatrix}