hw2 done up to ex5

hw2/hw2.tex

@@ -87,4 +87,67 @@ $$det(C - \lambda i) = 0 \lra = \lambda^2 - 16.25 \lambda + 42.25 = 0 \lra
 \lambda = 3.25 \lor \lambda = 13$$
 
 $$||A||_2 = \sqrt{\lambda_{max}(AA^T)} = \sqrt{13} \approx 3.606$$
+
+\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}