Q3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a b c 18

Q3

Q3.pdf

1

det(A)=∏i=0λidet(A)= \prod_{i=0} \lambda_i

2

A=QΣQ−1AT=(QΣQ−1)T=(QΣQ−1)T=(QTΣQ)\begin{aligned} A&=Q\Sigma Q^{-1}\\ A^T&=(Q\Sigma Q^{-1})^T\\ &=(Q\Sigma Q^{-1})^T\\ &=(Q^T\Sigma Q)\\ \end{aligned}

3

Ax=v1+v2x=v1+12v2v0 is in A nullspace Ax=v_1+v_2\\ x=v_1+\frac{1}{2}v_2\\ v_0 \text{ is in } A \text{ nullspace}

4

det(A1)=0det(A2)=−4det(A3)=4A1 cant diagonalizeddet(A_1)=0\\ det(A_2)=-4\\ det(A_3)=4\\ A_1\text{ cant diagonalized}

5

a.A2=IAv=λvA2v=vλ2=1b.det(A)=±1trace(A)=c.A=[3−18−3]\text{a.}\\ A^2=I\\ % A=Q\Sigma Q^{-1}\\ Av=\lambda v\\ A^2v=v\\ \lambda^2=1\\ \text{b.}\\ det(A)= \pm 1\\ trace(A)=\\ \text{c.}\\ A=\begin{bmatrix} 3 &-1 \\ 8 &-3 \\ \end{bmatrix}\\

6

A=[1111]P=A(ATA)−1ATA=\begin{bmatrix} 1 &1 &1 &1 \\ \end{bmatrix}\\ P=A(A^TA)^{-1}A^T

7

[Gk+1Gk+2]=[011212][GkGk+1]\begin{bmatrix} G_{k+1} \\ G_{k+2} \\ \end{bmatrix}= \begin{bmatrix} 0 &1\\ \frac{1}{2} &\frac{1}{2}\\ \end{bmatrix} \begin{bmatrix} G_k \\ G_{k+1} \\ \end{bmatrix}

8

A=QΣQ−1A=[.4.2.6.8]det(A−λI)=0(0.4−λ)(0.8−λ)−0.12=00.32−1.2λ+λ2−0.12=0λ2−1.2λ+0.2=0λ0=0.2,λ1=1Av0=λ0v0Av1=λ1v1Σ=[0.2001]Q=[−111.3]Ak=QΣkQ−1A=Q\Sigma Q^{-1}\\ A=\begin{bmatrix} .4 &.2 \\ .6 &.8 \\ \end{bmatrix}\\ det(A-\lambda I)=0\\ \begin{aligned}\\ (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{aligned}\\ \lambda_0=0.2 ,\lambda_1=1\\ Av_0=\lambda_0 v_0 \\ Av_1=\lambda_1 v_1 \\ \Sigma=\begin{bmatrix} 0.2 &0 \\ 0 &1 \\ \end{bmatrix}\\ Q=\begin{bmatrix} -1 &1 \\ 1 &.3 \\ \end{bmatrix}\\ A^k=Q\Sigma^k Q^{-1}\\

9

A=[1i0i01]AH=[1−i−i001]C=AHAC=[011]CH=CTyesA=\begin{bmatrix} 1 &i &0 \\ i &0 &1 \\ \end{bmatrix}\\ A^H=\begin{bmatrix} 1 & -i \\ -i &0 \\ 0 &1 \\ \end{bmatrix}\\ C=A^H A\\ C=\begin{bmatrix} 0 \\ 1 \\ 1 \\ \end{bmatrix}\\ C^H=C^T\\ yes

10

det(A−λI)=0(A−λ1I)v1=0(A−λ2I)v2=0A=λ1v1v1H+λ2v2v2Hdet(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

11

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

12

K=[iiii]KH=−K=[−i−i−i−i]det(K−λI)=0det([i−λiii−λ])=0λ1=2i,λ2=0K=SΣS−1K=[111−1][2i000][111−1]−1\begin{aligned} K&=\begin{bmatrix} i & i \\ i & i \\ \end{bmatrix}\\ K^H&=-K=\begin{bmatrix} -i & -i \\ -i & -i \\ \end{bmatrix}\\ \end{aligned}\\ det(K-\lambda I)= 0\\ det(\begin{bmatrix} i-\lambda & i \\ i & i-\lambda \\ \end{bmatrix})=0\\ \lambda_1 =2i ,\lambda_2=0\\ K=S \Sigma S^{-1}\\ K=\begin{bmatrix} 1 & 1 \\ 1 &-1 \\ \end{bmatrix} \begin{bmatrix} 2i & 0 \\ 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 1 &-1 \\ \end{bmatrix}^{-1}\\

13

A=(Z+ZH)÷2Z=A+KK=Z−AK=Z−(Z+ZH)÷2K=Z−ZH2A=(Z+Z^H)\div 2\\ Z=A+K\\ K=Z-A\\ K=Z-(Z+Z^H)\div 2\\ K=\frac{Z-Z^H}{2}

14

A=13[11−i1+i−1]det(A−λI)=0(A−λI)V=λVA=VΣVHA= \frac{1}{\sqrt{3} } \begin{bmatrix} 1 & 1-i \\ 1+i & -1 \\ \end{bmatrix}\\ det(A-\lambda I)=0\\ (A-\lambda I)V=\lambda V\\ A=V\Sigma V^H

15

V1=v1+v2V2=v2[10−11]V_1=v_1+v_2\\ V_2=v_2\\ \begin{bmatrix} 1 & 0 \\ -1 & 1 \\ \end{bmatrix}\\

16

A=[5445][1001]A=[5445][10−4/51]A=[5409/5]A=[10−4/51]−1[5409/5]A=[104/51][1/5005/9]A=LUA=[104/51][1/5005/9][14/501]A=LDUA= \begin{bmatrix} 5& 4 \\ 4 & 5 \\ \end{bmatrix}\\ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} A = \begin{bmatrix} 5& 4 \\ 4 & 5 \\ \end{bmatrix}\\ \begin{bmatrix} 1 & 0 \\ -4/5 & 1 \\ \end{bmatrix} A = \begin{bmatrix} 5& 4 \\ 0 & 9/5 \\ \end{bmatrix}\\ A = \begin{bmatrix} 1 & 0 \\ -4/5 & 1 \\ \end{bmatrix}^{-1} \begin{bmatrix} 5& 4 \\ 0 & 9/5 \\ \end{bmatrix}\\ A = \begin{bmatrix} 1 & 0 \\ 4/5 & 1 \\ \end{bmatrix} \begin{bmatrix} 1/5 & 0 \\ 0 & 5/9 \\ \end{bmatrix}\\ A=LU\\ A=\begin{bmatrix} 1 & 0 \\ 4/5 & 1 \\ \end{bmatrix} \begin{bmatrix} 1/5 & 0 \\ 0 & 5/9 \\ \end{bmatrix} \begin{bmatrix} 1& 4/5 \\ 0 & 1 \\ \end{bmatrix}\\ A=LDU\\

a

A=[xy][104/51][1/5005/9][14/501][xy]A=\begin{bmatrix} x & y \\ \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 4/5 & 1 \\ \end{bmatrix} \begin{bmatrix} 1/5 & 0 \\ 0 & 5/9 \\ \end{bmatrix} \begin{bmatrix} 1& 4/5 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix}\\

b

c

18

A=110[10608]A=UΣVTA=\frac{1}{\sqrt{10}}\begin{bmatrix} 10 & 6 \\ 0 & 8 \\ \end{bmatrix}\\ A=U\Sigma V^T\\