1. 

1. (a) ${\mathbf{u}}_1=\left[\begin{array}{c}1\\ 1\end{array}\right];$

2. (b) $A_2=\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right];$

3. (c) ${\lambda }_1=2,{\lambda }_2=0$; the eigenspace corresponding to ${\lambda }_1$ is spanned by ${\mathbf{u}}_1$

2. 

1. (a)
   $\begin{array}{cccc}{\mathbf{v}}_1& =\left[\begin{array}{c}3\\ 5\\ 3\end{array}\right],& {\mathbf{u}}_1& =\left[\begin{array}{c}0.6\\ 1.0\\ 0.6\end{array}\right],\\ {\mathbf{v}}_2& =\left[\begin{array}{c}2.2\\ 4.2\\ 2.2\end{array}\right],& {\mathbf{u}}_2& =\left[\begin{array}{c}0.52\\ 1.00\\ 0.52\end{array}\right],\\ {\mathbf{v}}_3& =\left[\begin{array}{c}2.05\\ 4.05\\ 2.05\end{array}\right]\mathrm{;}\end{array}$

2. (b) ${\lambda }_1^{\prime }=4.05;$

3. (c) ${\lambda }_1=4,\delta =0.0125$

3. Hint: How are the eigenvalues of A related?

4. $\begin{array}{c}\begin{array}{cc}{A}_2=\left[\begin{array}{cc}\phantom{-}3& -1\\ -1& \phantom{-}1\end{array}\right],& A_3=\left[\begin{array}{cc}3.4& 0.2\\ 0.2& 0.6\end{array}\right],\end{array}\hfill \\ {\lambda }_1=2=\sqrt{2}\approx 3.141,{\lambda }_2=2-\sqrt{2}\approx 0.586\end{array}$

5. 

1. (b) $H=I-{\displaystyle \frac{1}{\beta }}{\mathbf{\text{vv}}}^T$, where $\beta =\frac{1}{3}$ and $\mathbf{v}=(-\frac{1}{3},-\frac{2}{3},\frac{1}{3}{)}^T$;

2. (c) ${\lambda }_2=3,{\lambda }_3=1,HAH=\left[\begin{array}{ccc}4& 0& 3\\ 0& 5& -4\\ 0& 2& -1\end{array}\right]$

6.  

1. (a) Hint: If ${\mathbf{\text{x}}}_j$ is an eigenvector of A belonging to ${\lambda }_j$, then

   $$B^{-1}{\mathbf{\text{x}}}_j=(A-\lambda I){\mathbf{\text{x}}}_j=({\lambda }_j-\lambda ){\mathbf{\text{x}}}_j=\frac{1}{{\mu }_j}{\mathbf{\text{x}}}_j$$

7.  

1. (b) Hint: It follows from part (a) that

   $$\left(\lambda -a_{ii}\right)x_i={\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^n{a}_{ij}{x}_j}$$

8. Hint: A $A^T$ have the same eigenvalues.