\section*{Problem 2}
For a square matrix $\boldsymbol { A } , e ^ { \boldsymbol { A } }$ is defined as:

$$e ^ { A } = \boldsymbol { E } + \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ! } \boldsymbol { A } ^ { k } ,$$

where $\boldsymbol { E }$ is the identity matrix and $e$ is the base of natural logarithm.

I. Let $\boldsymbol { A }$ be a $3 \times 3$ square matrix which can be diagonalized by a regular matrix $\boldsymbol { P }$, i.e., $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$, where $\boldsymbol { D }$ is a diagonal matrix:

$$\boldsymbol { D } = \left( \begin{array} { c c c } 
\lambda _ { 1 } & 0 & 0 \\
0 & \lambda _ { 2 } & 0 \\
0 & 0 & \lambda _ { 3 }
\end{array} \right)$$

Here, $\lambda _ { 1 } , \lambda _ { 2 }$, and $\lambda _ { 3 }$ are complex numbers. Prove the following equation:

$$e ^ { \boldsymbol { A } } = \boldsymbol { P } \left( \begin{array} { c c c } 
e ^ { \lambda _ { 1 } } & 0 & 0 \\
0 & e ^ { \lambda _ { 2 } } & 0 \\
0 & 0 & e ^ { \lambda _ { 3 } }
\end{array} \right) \boldsymbol { P } ^ { - 1 }$$

II. Let $\boldsymbol { A } = \left( \begin{array} { c c c } - 1 & 4 & 4 \\ - 5 & 8 & 10 \\ 3 & - 3 & - 5 \end{array} \right)$.

\begin{enumerate}
  \item Find the regular matrix $\boldsymbol { P }$ and the diagonal matrix $\boldsymbol { D }$ such that $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$.
  \item Calculate $e ^ { \boldsymbol { A } }$.
\end{enumerate}

III. Consider $\boldsymbol { A } = \left( \begin{array} { c c c } 0 & - x & 0 \\ x & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) , \boldsymbol { B } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$ and $\boldsymbol { a } = \left( \begin{array} { l } 1 \\ 1 \\ e \end{array} \right)$, where $x$ is a real number. In the following, the transpose of a vector $\boldsymbol { v }$ is denoted by $\boldsymbol { v } ^ { T }$.

\begin{enumerate}
  \item Express the sum of the eigenvalues of $e ^ { \boldsymbol { A } }$ using $e$ and $x$.
  \item Let $\boldsymbol { C } = \boldsymbol { B } e ^ { \boldsymbol { A } }$. Find the minimum and maximum values of $\frac { \boldsymbol { y } ^ { T } \boldsymbol { C } \boldsymbol { y } } { \boldsymbol { y } ^ { T } \boldsymbol { y } }$ for a real three-dimensional vector $\boldsymbol { y } ( \boldsymbol { y } \neq \mathbf { 0 } )$.
  \item Let $f ( \boldsymbol { z } ) = \frac { 1 } { 2 } \boldsymbol { z } ^ { T } \boldsymbol { C } \boldsymbol { z } - \boldsymbol { a } ^ { T } \boldsymbol { z }$ for a real three-dimensional vector $\boldsymbol { z } = \left( \begin{array} { c } z _ { 1 } \\ z _ { 2 } \\ z _ { 3 } \end{array} \right)$ and $\boldsymbol { C }$ in III.2. Find $\sqrt { z _ { 1 } ^ { 2 } + z _ { 2 } ^ { 2 } + z _ { 3 } ^ { 2 } }$ for $\boldsymbol { z }$ such that $\frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 1 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 2 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 3 } } = 0$.
\end{enumerate}