Q3

Q3.pdf

1

$$det(A)=\prod _{i=0}{\lambda }_i$$

2

$$\begin{array}{rl}A& =Q\Sigma Q^{-1}\\ A^T& =(Q\Sigma Q^{-1}{)}^T\\ & =(Q\Sigma Q^{-1}{)}^T\\ & =(Q^T\Sigma Q)\end{array}$$

3

$$\begin{gathered} Ax=v_1+v_2 \\ x=v_1+\frac{1}{2}{v}_2 \\ {v}_0\text{ is in }A\text{ nullspace} \end{gathered}$$

4

$$\begin{gathered} det(A_1)=0 \\ det(A_2)=-4 \\ det(A_3)=4 \\ {A}_1\text{ cant diagonalized} \end{gathered}$$

5

$$\begin{gathered} \text{a.} \\ {A}^2=I \\ Av=\lambda v \\ {A}^{2}v=v \\ {\lambda }^2=1 \\ \text{b.} \\ det(A)=\pm 1 \\ trace(A)= \\ \text{c.} \\ A=\left[\begin{array}{cc}3& -1\\ 8& -3\end{array}\right] \end{gathered}$$

6

$$\begin{gathered} A=\left[\begin{array}{cccc}1& 1& 1& 1\end{array}\right] \\ P=A(A^{T}A{)}^{-1}{A}^T \end{gathered}$$

7

$$\left[\begin{array}{c}{G}_{k+1}\\ G_{k+2}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}{G}_k\\ G_{k+1}\end{array}\right]$$

8

$$\begin{gathered} A=Q\Sigma Q^{-1} \\ A=\left[\begin{array}{cc}.4& .2\\ .6& .8\end{array}\right] \\ det(A-\lambda I)=0 \\ \begin{array}{rl}& \\ (0.4-\lambda )(0.8-\lambda )-0.12& =0\\ 0.32-1.2\lambda +{\lambda }^2-0.12& =0\\ {\lambda }^2-1.2\lambda +0.2& =0\end{array} \\ {\lambda }_0=0.2,{\lambda }_1=1 \\ A{v}_0={\lambda }_0{v}_0 \\ A{v}_1={\lambda }_1{v}_1 \\ \Sigma =\left[\begin{array}{cc}0.2& 0\\ 0& 1\end{array}\right] \\ Q=\left[\begin{array}{cc}-1& 1\\ 1& .3\end{array}\right] \\ {A}^k=Q{\Sigma }^k{Q}^{-1} \end{gathered}$$

9

$$\begin{gathered} A=\left[\begin{array}{ccc}1& i& 0\\ i& 0& 1\end{array}\right] \\ {A}^H=\left[\begin{array}{cc}1& -i\\ -i& 0\\ 0& 1\end{array}\right] \\ C=A^{H}A \\ C=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right] \\ {C}^H=C^T \\ yes \end{gathered}$$

10

$$\begin{gathered} det(A-\lambda I)=0 \\ (A-{\lambda }_{1}I)v_1=0 \\ (A-{\lambda }_{2}I)v_2=0 \\ A={\lambda }_1{v}_1{v}_1^H+{\lambda }_2{v}_2{v}_2^H \end{gathered}$$

11

• (a) real eigenvalue
• (b) real part < 0
• (c) eigenvalue 1 or -1
• (d) eigenvalue =1
• (e)
• (f) eigenvalue =0

12

$$\begin{gathered} \begin{array}{rl}K& =\left[\begin{array}{cc}i& i\\ i& i\end{array}\right]\\ K^H& =-K=\left[\begin{array}{cc}-i& -i\\ -i& -i\end{array}\right]\end{array} \\ det(K-\lambda I)=0 \\ det(\left[\begin{array}{cc}i-\lambda & i\\ i& i-\lambda \end{array}\right])=0 \\ {\lambda }_1=2i,{\lambda }_2=0 \\ K=S\Sigma S^{-1} \\ K=\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\left[\begin{array}{cc}2i& 0\\ 0& 0\end{array}\right]{\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]}^{-1} \end{gathered}$$

13

$$\begin{gathered} A=(Z+Z^H)÷2 \\ Z=A+K \\ K=Z-A \\ K=Z-(Z+Z^H)÷2 \\ K=\frac{Z-Z^H}{2} \end{gathered}$$

14

$$\begin{gathered} A=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1& 1-i\\ 1+i& -1\end{array}\right] \\ det(A-\lambda I)=0 \\ (A-\lambda I)V=\lambda V \\ A=V\Sigma V^H \end{gathered}$$

15

$$\begin{gathered} V_1=v_1+v_2 \\ {V}_2=v_2 \\ \left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right] \end{gathered}$$

16

$$\begin{gathered} A=\left[\begin{array}{cc}5& 4\\ 4& 5\end{array}\right] \\ \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]A=\left[\begin{array}{cc}5& 4\\ 4& 5\end{array}\right] \\ \left[\begin{array}{cc}1& 0\\ -4/5& 1\end{array}\right]A=\left[\begin{array}{cc}5& 4\\ 0& 9/5\end{array}\right] \\ A={\left[\begin{array}{cc}1& 0\\ -4/5& 1\end{array}\right]}^{-1}\left[\begin{array}{cc}5& 4\\ 0& 9/5\end{array}\right] \\ A=\left[\begin{array}{cc}1& 0\\ 4/5& 1\end{array}\right]\left[\begin{array}{cc}1/5& 0\\ 0& 5/9\end{array}\right] \\ A=LU \\ A=\left[\begin{array}{cc}1& 0\\ 4/5& 1\end{array}\right]\left[\begin{array}{cc}1/5& 0\\ 0& 5/9\end{array}\right]\left[\begin{array}{cc}1& 4/5\\ 0& 1\end{array}\right] \\ A=LDU \end{gathered}$$

a

$$A=\left[\begin{array}{cc}x& y\end{array}\right]\left[\begin{array}{cc}1& 0\\ 4/5& 1\end{array}\right]\left[\begin{array}{cc}1/5& 0\\ 0& 5/9\end{array}\right]\left[\begin{array}{cc}1& 4/5\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

b

c

18

$$\begin{gathered} A=\frac{1}{\sqrt{10}}\left[\begin{array}{cc}10& 6\\ 0& 8\end{array}\right] \\ A=U\Sigma V^T \end{gathered}$$