Consider the following matrix $\boldsymbol { A }$ :
$$A = \left( \begin{array} { c c c }
1 & - 2 & - 1 \\
- 2 & 1 & 1 \\
- 1 & 1 & \alpha
\end{array} \right)$$
where $\alpha$ is a real number. In the following, the transpose of a vector $\boldsymbol { v }$ is denoted by $\boldsymbol { v } ^ { \mathrm { T } }$.
I. Obtain $\alpha$ when the sum of the three eigenvalues of the matrix $A$ is 7.
II. Obtain $\alpha$ when the product of the three eigenvalues of the matrix $\boldsymbol { A }$ is $- 16$.
III. Let $\| \boldsymbol { A } \|$ be the maximum of $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A x }$ for the set of real vectors $\boldsymbol { x } = \left( \begin{array} { l } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{array} \right)$ that satisfy $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { x } = 1$. Obtain $\alpha$ when $\| \boldsymbol { A } \| = 4$.
IV. In the following questions, $\alpha = 4$.
- Obtain all eigenvalues of the matrix $\boldsymbol { A }$ and their corresponding normalized eigenvectors.
- Find the range of $\boldsymbol { y } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { y }$ for the real vectors $\boldsymbol { y } = \left( \begin{array} { l } y _ { 1 } \\ y _ { 2 } \\ y _ { 3 } \end{array} \right)$ that satisfy $\boldsymbol { y } ^ { \mathrm { T } } \boldsymbol { y } = 1$ and $y _ { 1 } - y _ { 2 } - 2 y _ { 3 } = 0$.
- Find the range of $\boldsymbol { z } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { z }$ for the real vectors $\boldsymbol { z } = \left( \begin{array} { c } z _ { 1 } \\ z _ { 2 } \\ z _ { 3 } \end{array} \right)$ that satisfy $\boldsymbol { z } ^ { \mathrm { T } } \boldsymbol { z } = 1$ and $z _ { 1 } + z _ { 2 } + z _ { 3 } = 0$.