From d897a331eb5a5b7addcdb13184e33d670dcc13ac Mon Sep 17 00:00:00 2001 From: "Claudio Maggioni (maggicl)" Date: Thu, 23 Apr 2020 15:58:53 +0200 Subject: [PATCH] hw3: done 1,2,3,4,5a,6 --- hw3/hw3.pdf | Bin 72320 -> 89252 bytes hw3/hw3.tex | 31 +++++++++++++++++++++++++++++++ 2 files changed, 31 insertions(+) diff --git a/hw3/hw3.pdf b/hw3/hw3.pdf index 7491c8ce89807bb19e31e63302eeeb8cb1cf9935..5f804a99d8bac8226eb31f552a4b647f3d5c0e5e 100644 GIT binary patch delta 73772 zcmZtNQ*fY76ei$gVmp~66K7)E#>BR5n_ujSZQHhOV`AI({<~XSd$D`h*InC`VY$4UlI7}hW?o5p7!jzmG`xQP4{olcoFfmV@6uj;~ie{lX!6h*BP&-6qoc^ zG1=Wrw&1zMao9TAf1@79LpknLshwHc-MQb)cyd0xZr2yob~2K6iB{vLG3m*IZw;*L z4ZI}J6PcB@0N+w+8ad>y5M%#wNPa4T*RZwmKh%&#FyaUmSZlB{3$Ri37a##l?VPyW z@;~MO3E?|G=Bn*Dm`xU~&8H@@6Wm|#a7p>8EVmCAZ1xdy%23@bk5XJ$m?+$>!SQ+Z znZ>(P5L)`uM8QHu2#prgL-76R5KKc_zcJ+}a5d+v1bU%W82I22dq zRP%omv!dP8+uEZ4f>|ASSy|2Jgyqjjyy2!cMZHb7w|`h)Kh#3Y3;+6>K9ydA4%OrDPdNu<*K|k|1~W`^ZGZT7~wQRfhjvlMwGGmk2ZcV 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b/hw3/hw3.tex @@ -255,4 +255,35 @@ A_2 = \begin{bmatrix} \[L = \begin{bmatrix}1\\4&2\\8&4&3\\3&8&5&4\\\end{bmatrix}\] +\section*{Question 5} + +\subsection*{Point a)} +First of all, to show that $A_{1,1}$ is symmetric, we say: + +\[\begin{bmatrix}A_{1,1} & A_{1,2}\\A_{2,1} & A_{2,2}\\\end{bmatrix} = A = A^T = \begin{bmatrix}A_{1,1}^T & A_{2,1}\\A_{1,2} & A_{2,2}^T\\\end{bmatrix}\] + +Therefore we can say that + +$$A_{1,1} = A_{1,1}^T$$ + +and thus $A_{1,1}$ is shown to be symmetric. + +Then, since $A$ is positive definite, $x^TAx > 0$ for any vector $x \in R^n$. We choose a vector $x$ such that $x_k = 0$ for any $p < k \leq n$. + +Then: + +$$ x^TAx = \sum_{j=1}^n x_j \cdot \sum_{i=1}^n x_i a_{i,j} = \sum_{j=1}^p x_j \cdot \sum_{i=1}^n x_i a_{i,j} + +\sum_{j=p+1}^n 0 \cdot \sum_{i=1}^n 0 a_{i,j} = +\sum_{j=1}^p x_j \cdot (\sum_{i=i}^p x_i a_{i,j} + +\sum_{i=p+1}^n 0 a_{i,j}) =$$ +$$=\sum_{j=1}^p x_j \cdot \sum_{i=i}^p x_i a_{i,j} = +\begin{bmatrix}x_1 & x_2 & \ldots & x_n\end{bmatrix} +A_{1,1}\begin{bmatrix}x_1 & x_2 & \ldots & x_n\end{bmatrix}^T$$ + +Then: + +$$\begin{bmatrix}x_1 & x_2 & \ldots & x_n\end{bmatrix}A_{1,1}\begin{bmatrix}x_1 & x_2 & \ldots & x_n\end{bmatrix}^T>0$$ + +which is the definition of positive definiteness for $A_{1,1}$. + \end{document}