Consider expressing the following matrix $\boldsymbol { A }$ in a form of $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$, using a diagonal matrix $\boldsymbol { D }$ and a regular matrix $\boldsymbol { P }$. Here, $a$ is a real number. $$A = \left( \begin{array} { l l l }
2 & 1 & 0 \\
1 & 3 & a \\
0 & a & 2
\end{array} \right)$$ I. When $a = 1$, find a diagonal matrix $D$. II. When $a = 1$, prove $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } > 0$ for any three-dimensional non-zero real vector $\boldsymbol { x }$. $\boldsymbol { x } ^ { \mathrm { T } }$ represents the transpose of $\boldsymbol { x }$. III. Find the condition of $a$ which satisfies $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } > 0$ for any three-dimensional non-zero real vector $\boldsymbol { x }$. IV. Assume that $a$ satisfies the condition obtained in Question III. For a real vector $\boldsymbol { b } = \left( \begin{array} { c } a \\ 0 \\ - 1 \end{array} \right)$, express the minimum value of the function $f ( \boldsymbol { x } ) = \boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } - \boldsymbol { b } ^ { \mathrm { T } } \boldsymbol { x }$ by using $a$.
Consider expressing the following matrix $\boldsymbol { A }$ in a form of $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$, using a diagonal matrix $\boldsymbol { D }$ and a regular matrix $\boldsymbol { P }$. Here, $a$ is a real number.
$$A = \left( \begin{array} { l l l }
2 & 1 & 0 \\
1 & 3 & a \\
0 & a & 2
\end{array} \right)$$
I. When $a = 1$, find a diagonal matrix $D$.
II. When $a = 1$, prove $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } > 0$ for any three-dimensional non-zero real vector $\boldsymbol { x }$. $\boldsymbol { x } ^ { \mathrm { T } }$ represents the transpose of $\boldsymbol { x }$.
III. Find the condition of $a$ which satisfies $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } > 0$ for any three-dimensional non-zero real vector $\boldsymbol { x }$.
IV. Assume that $a$ satisfies the condition obtained in Question III.
For a real vector $\boldsymbol { b } = \left( \begin{array} { c } a \\ 0 \\ - 1 \end{array} \right)$, express the minimum value of the function $f ( \boldsymbol { x } ) = \boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } - \boldsymbol { b } ^ { \mathrm { T } } \boldsymbol { x }$ by using $a$.