We define the matrix $W_n = (w_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$ by $w_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } i = n \text{ and } j \in \{1,2\} \\ 0 & \text{in all other cases} \end{cases}$
Show that the characteristic polynomial of $W_n$ is $X^n - X - 1$.
Deduce that $W_n^{n^2-2n+1} = \sum_{k=1}^{n-1} \binom{n-2}{k-1} W_n^k$, then that $W_n^{n^2-2n+2} = I_n + W_n + \sum_{k=2}^{n-1} \binom{n-2}{k-2} W_n^k$.

Let $A$ and $b$ be defined as
$$A = \left( \begin{array} { r r r } - 3 & 0 & 0 \\ - 2 & - 3 & 1 \\ 2 & - 3 & - 3 \end{array} \right) , b = \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) .$$
The partial derivative of a scalar-valued function $f ( x )$ with respect to $x = \left( \begin{array} { l l l } x _ { 1 } & x _ { 2 } & x _ { 3 } \end{array} \right) ^ { T }$ is defined as
$$\frac { \partial } { \partial x } f ( x ) = \left( \frac { \partial } { \partial x _ { 1 } } f ( x ) \quad \frac { \partial } { \partial x _ { 2 } } f ( x ) \quad \frac { \partial } { \partial x _ { 3 } } f ( x ) \right)$$
and a stationary point of $f ( x )$ is defined as $x$ satisfying $\frac { \partial } { \partial x } f ( x ) = \left( \begin{array} { l l l } 0 & 0 & 0 \end{array} \right) . x ^ { T }$ denotes the transpose of $x$. Answer the following questions.
(1) Find the characteristic polynomial of $A$.
(2) $C$ is given as $C = A ^ { 5 } + 9 A ^ { 4 } + 30 A ^ { 3 } + 36 A ^ { 2 } + 30 A + 9 I$ by using $A$ and an identity matrix $I$. Calculate $C$.
(3) Calculate the partial derivative of $x ^ { T } A x$ with respect to $x$.
(4) Find a symmetric matrix $\tilde { A }$ that satisfies equation $x ^ { T } A x = x ^ { T } \tilde { A } x$ for any vector $x$. Find eigenvalues $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \left( \lambda _ { 1 } \geq \lambda _ { 2 } \geq \lambda _ { 3 } \right)$, and eigenvectors $v _ { 1 } , v _ { 2 } , v _ { 3 }$. Choose the eigenvectors such that $V = \left( v _ { 1 } v _ { 2 } v _ { 3 } \right)$ becomes an orthogonal matrix.
(5) Prove that $x ^ { T } A x \leq 0$ holds for any real vector $x$.
(6) Find a stationary point of function $g ( x ) = x ^ { T } A x + 2 b ^ { T } x$.