\textbf{Problem 2}

Consider the column vectors $\mathbf { a } _ { 1 } = \left( \begin{array} { l } 0 \\ 1 \\ 1 \end{array} \right) , \mathbf { a } _ { 2 } = \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) , \mathbf { a } _ { 3 } = \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 1 \\ 2 \\ 4 \end{array} \right) , \mathbf { 0 } = \left( \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right)$.

I. When $\mathbf { A } = \left( \begin{array} { l l l } \mathbf { a } _ { 1 } & \mathbf { a } _ { 2 } & \mathbf { a } _ { 3 } \end{array} \right)$, obtain the three-dimensional column vector $\mathbf { x }$ which meets

$$A x - b = 0 .$$

II. Any $m \times n$ real matrix $\mathbf { B }$ is expressed using orthonormal matrices $\mathbf { U } ( m \times m )$ and $\mathrm { V } ( n \times n )$ as

$$\mathbf { B } = \mathbf { U \Sigma V } ^ { T } , \quad \boldsymbol { \Sigma } = \left( \begin{array} { c c c c c c c } 
\sigma _ { 1 } & 0 & \cdots & 0 & 0 & \cdots & 0 \\
0 & \sigma _ { 2 } & \ddots & \vdots & \vdots & & \vdots \\
\vdots & \ddots & \ddots & 0 & \vdots & & \vdots \\
0 & \cdots & 0 & \sigma _ { r } & 0 & \cdots & 0 \\
0 & \cdots & \cdots & 0 & 0 & \cdots & 0 \\
\vdots & & & \vdots & \vdots & & \vdots \\
0 & \cdots & \cdots & 0 & 0 & \cdots & 0
\end{array} \right) , \quad r = \operatorname { rank } ( \mathbf { B } ) .$$

$\sigma _ { 1 } , \sigma _ { 2 } , \cdots , \sigma _ { r }$ are positive real numbers, and they are called singular values of $\mathbf { B }$. $\mathbf { P } ^ { T }$ means the transposed matrix of a matrix $\mathbf { P }$. Then, express $\mathbf { B B } ^ { T }$ and $\mathbf { B } ^ { T } \mathbf { B }$ using matrices $\mathbf { U } , \mathbf { V } , \boldsymbol { \Sigma }$ and their transposed matrices, respectively.

Let $\mathbf { B } = \left( \mathbf { a } _ { 1 } \mathbf { a } _ { 2 } \right)$ in the following questions.

III. Find the eigenvalues and corresponding eigenvectors for $\mathbf { B B } ^ { T }$.

IV. Find singular values of $\mathbf { B }$ and orthonormal matrices $\mathbf { U }$ and $\mathbf { V }$ used in Equation (2).

V. Find the two-dimensional column vector $\mathbf { x }$ which minimizes the norm

$$\| \mathrm { Bx } - \mathrm { b } \| ^ { 2 } = ( \mathrm { Bx } - \mathrm { b } ) ^ { T } ( \mathrm { Bx } - \mathrm { b } ) .$$