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\lambda = 13$$ $$||A||_2 = \sqrt{\lambda_{max}(AA^T)} = \sqrt{13} \approx 3.606$$ + +\section*{Exercise 5} +\subsection*{Point a} +\[ + A^{-1} = + \frac{1}{(-2) - 4.5} + \begin{bmatrix} + -1 & -3 \\ + -1.5 & 2 \\ + \end{bmatrix} + = + \begin{bmatrix} + 2/13 & 6/13 \\ + 3/13 & -4/13 \\ + \end{bmatrix} + \] + +$$||A^{-1}||_\infty = \max_{1 \leq j \leq 2}{\sum_{j=1}^{2} |a_{i,j}|} = + \max{\{\frac{8}{13}. \frac{7}{13}\}} = \frac{8}{13}$$ + +$$||A^{-1}||_1 = \max_{1 \leq i \leq 2}{\sum_{i=1}^{2} |a_{i,j}|} = + \max{\{\frac{5}{13}, \frac{10}{13}\}} = \frac{10}{13}$$ + +$$||A^{-1}||_F = \left(\sum_{i=1}^{2}\sum_{j=1}^{2} +(a_{i,j})^2\right)^{\frac{1}{2}} = \left(\frac{65}{13^2}\right)^{\frac{1}{2}} += \sqrt{\frac{5}{13}}$$ + +$$k_1(A) = \frac{4 * 10}{13} \approx 3.077$$ + +$$k_\infty(A) = \frac{5 * 8}{13} \approx 3.077$$ + +$$k_F(A) \approx 8.660 * \sqrt{\frac{5}{13}} \approx 5.371$$ + +\subsection*{Point b} +$$||A^{-1}||_2 = \sigma_{max}(A^{-1}) = \frac{1}{\sigma_{min}(A)} = \frac{1}{\sqrt{\lambda_{min}}} =\frac{1}{\sqrt{3.25}}$$ + +$$k_2(A) = \frac{\sqrt{13}}{\sqrt{3.25}} = 2$$ + +\subsection*{Point c} + +\[ + k_\infty + \begin{bmatrix} + -4 & 0 & 0 \\ + 0 & 5 & 0 \\ + 0 & 0 & 3 \\ + \end{bmatrix} + = + \left|\left| \begin{bmatrix} + -4 & 0 & 0 \\ + 0 & 5 & 0 \\ + 0 & 0 & 3 \\ + \end{bmatrix} \right|\right|_\infty + \left|\left| \begin{bmatrix} + -1/4 & 0 & 0 \\ + 0 & 1/5 & 0 \\ + 0 & 0 & 1/3 \\ + \end{bmatrix} \right|\right|_\infty + = 5 * \frac{1}{3} = \frac{5}{3} +\] + +\subsection*{Point d} +The condition of that matrix cannot be computed since that matrix has no inverse since it is not full rank ($M_{2,:} = M_{1,:} * -2$). \end{document}