1. Problem (a): Find the SVD of matrix $$A = \begin{pmatrix} -2 & 2 \\ 1 & 1 \end{pmatrix}$$. 2. Step 1: Compute $$A^TA$$ and $$AA^T$$. $$A^TA = \begin{pmatrix} -2 & 1 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} -2 & 2 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 5 & -3 \\ -3 & 5 \end{pmatrix}$$ $$AA^T = \begin{pmatrix} -2 & 2 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} -2 & 1 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 8 & 0 \\ 0 & 2 \end{pmatrix}$$ 3. Step 2: Find eigenvalues and eigenvectors of $$A^TA$$ and $$AA^T$$. - For $$A^TA$$: Solve $$\det(A^TA - \lambda I) = 0$$: $$\det\left(\begin{pmatrix} 5-\lambda & -3 \\ -3 & 5-\lambda \end{pmatrix}\right) = (5-\lambda)^2 - 9 = 0$$ $$\Rightarrow (5-\lambda)^2 = 9 \Rightarrow 5-\lambda = \pm 3$$ So eigenvalues: $$\lambda_1 = 2$$ and $$\lambda_2 = 8$$. - For $$\lambda=8$$, eigenvector solves: $$\begin{pmatrix} -3 & -3 \\ -3 & -3 \end{pmatrix}{\bf v} = 0$$ so $$v_1 = (1, -1)^T$$ (normalized). - For $$\lambda=2$$, eigenvector solves: $$\begin{pmatrix} 3 & -3 \\ -3 & 3 \end{pmatrix}{\bf v} = 0$$ so $$v_2 = (1, 1)^T$$ (normalized). - For $$AA^T$$: Diagonal matrix with eigenvalues $$8$$ and $$2$$ and eigenvectors $$u_1 = (1,0)^T$$ and $$u_2 = (0,1)^T$$. 4. Step 3: Singular values $$\sigma_i = \sqrt{\lambda_i}$$ are: $$\sigma_1 = \sqrt{8} = 2\sqrt{2}$$, $$\sigma_2 = \sqrt{2}$$. 5. Step 4: Form matrices for SVD: - $$U = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ (from eigenvectors of $$AA^T$$). - $$\Sigma = \begin{pmatrix} 2\sqrt{2} & 0 \\ 0 & \sqrt{2} \end{pmatrix}$$. - $$V = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$ (from eigenvectors of $$A^TA$$). 6. Problem (b): Given $$A=\begin{pmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{pmatrix}$$. (i) Compute $$A^TA$$ and $$AA^T$$: $$A^TA = \begin{pmatrix} 3 & 2 \\ 2 & 3 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{pmatrix} = \begin{pmatrix} 13 & 12 & 2 \\ 12 & 13 & -10 \\ 2 & -10 & 8 \end{pmatrix}$$ $$AA^T = \begin{pmatrix} 17 & 4 \\ 4 & 17 \end{pmatrix}$$ (ii) Find eigenvalues and eigenvectors of $$A^TA$$ and $$AA^T$$. - For $$AA^T$$: $$\det\left( \begin{pmatrix}17-\lambda & 4 \\ 4 & 17 - \lambda\end{pmatrix} \right) = (17-\lambda)^2 - 16 = 0$$ $$\Rightarrow (17-\lambda)^2 = 16 \Rightarrow 17-\lambda = \pm 4$$ Eigenvalues: $$\lambda_1=21$$, $$\lambda_2=13$$. - For $$A^TA$$: Find eigenvalues by solving $$\det(A^TA - \lambda I) = 0$$. This cubic characteristic polynomial is: $$- \lambda^3 + 34 \lambda^2 - 337 \lambda + 756 = 0$$ Eigenvalues match those of $$AA^T$$ and the extra eigenvalue 4. (iii) Singular values: $$\sigma_i = \sqrt{\lambda_i}$$ are $$\sqrt{21}$$, $$\sqrt{13}$$, and $$2$$. (iv) SVD is: $$ A = U \Sigma V^T $$ where $$U$$ has eigenvectors of $$AA^T$$, $$V$$ of $$A^TA$$, and $$\Sigma$$ diagonal with singular values. 7. Problem (c): Given points $$ (x,y): (1,0), (-1,1), (2,2), (0,1) $$. (a) Write system for polynomial $$p(x) = a_0 + a_1 x + a_2 x^2$$ interpolating points. Construct system: $$\begin{cases} a_0 + a_1 (1) + a_2 (1)^2 = 0 \\ a_0 + a_1 (-1) + a_2 (1) = 1 \\ a_0 + a_1 (2) + a_2 (4) = 2 \\ a_0 + a_1 (0) + a_2 (0) = 1 \end{cases}$$ (b) Solve using SVD or software to get coefficients $a_0, a_1, a_2$. Final answers: - (a) $$U= \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}, \Sigma=\begin{pmatrix}2\sqrt{2} & 0\\0 & \sqrt{2}\end{pmatrix}, V=\begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$. - (b)(i) $$A^TA = \begin{pmatrix}13 & 12 & 2 \\ 12 & 13 & -10 \\ 2 & -10 & 8 \end{pmatrix}, AA^T = \begin{pmatrix} 17 & 4 \\ 4 & 17 \end{pmatrix}$$. - (b)(iii) Singular values: $$\sqrt{21}, \sqrt{13}, 2$$. - (c)(a) System shown. - (c)(b) Solve for polynomial coefficients.