From b1e347f4c3cf04f14786f5f095a8e74d242e628f Mon Sep 17 00:00:00 2001 From: ersoycaliskan Date: Fri, 3 Mar 2017 18:47:22 +0300 Subject: [PATCH] HW3 is done --- Untitled.ipynb | 485 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 485 insertions(+) create mode 100644 Untitled.ipynb diff --git a/Untitled.ipynb b/Untitled.ipynb new file mode 100644 index 0000000..d9acdc6 --- /dev/null +++ b/Untitled.ipynb @@ -0,0 +1,485 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\\textbf{3.6 a)}$\n", + "\n", + "$||x||^{'}=\\underset{||y||=1}{sup}|y^{*}x|$\n", + "\n", + "$\\textit{Condition 1:}$ $\\underset{||y||=1}{sup}|y^{*}x| \\geq 0$ \n", + "\n", + "Let say $y=\\frac{x}{||x||}$ \n", + "\n", + "Therefore $||x||^{'} \\geq |(\\frac{x}{||x||})^{*}x| \\geq 0$ and it means that $||x||^{'} \\geq 0$\n", + "\n", + "$\\textit{Condition 2:}$ If $\\underset{||y||=1}{sup}|y^{*}x|=0$ then $x$ must be zero. \n", + "\n", + "It is obvious. Because if $y=\\frac{x}{||x||}$ then $||x||^{'} \\geq |(\\frac{x}{||x||})^{*}x| \\geq 0$\n", + "\n", + "If $|(\\frac{x}{||x||})^{*}x| = 0$ then $x$ must be zero. And if $x$ is zero $||x||^{'}$ is also zero.\n", + "\n", + "\n", + "$\\textit{Condition 3:}$ $||a+b||^{'} \\leq ||a||^{'}+||b||^{'}$\n", + "\n", + "$||a+b||^{'} =\\underset{||y||=1}{sup}|y^{*}(a+b)| \\leq \\underset{||y||=1}{sup}|y^{*}a|+|y^{*}b| \\leq \\underset{||y||=1}{sup}|y^{*}a| + \\underset{||y||=1}{sup}|y^{*}b| = ||a||^{'} + ||b||^{'}$\n", + "\n", + "$\\textit{Condition 4:}$ $||\\alpha x||^{'} = |\\alpha|.||x||^{'}$\n", + "\n", + "$||\\alpha x||^{'}=\\underset{||y||=1}{sup}|y^{*}(\\alpha x)| = \\underset{||y||=1}{sup}|\\alpha y^{*}x|=\\underset{||y||=1}{sup}|\\alpha|.|y^{*}x|=|\\alpha|.\\underset{||y||=1}{sup}|y^{*}x|=|\\alpha|.||x||^{'}$\n", + "\n", + "Therefore $||x||^{'}$ is a norm." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\\textbf{4.5}$\n", + "\n", + "SVD of $A$ is $U \\Sigma V^{*}$ where $\\Sigma$ is a diagonal matrix.\n", + "\n", + "Let say $A^{*}A$ is matrix $B$. Notice that $B$ is a real symmetric square matrix (i.e. $B=B^{*}$). Eigen values of the $B$ are diagonal elements of the $\\Sigma^{2}$ and Eigen vectors of the $B$ are columns of the $V$. \n", + "\n", + "Now we have to prove that eigen values and eigen vectors of a real symmetric matix are always real.\n", + "\n", + "Assume $a+ib$ is a eigenvalue of the matrix $B$ and its corresponding eigenvector is $\\textbf{v}+i \\textbf{w}$. In other words $B(\\textbf{v}+i \\textbf{w})=(a+ib)(\\textbf{v}+i \\textbf{w})$ \n", + "\n", + "Consider this eqution: $(\\textbf{v}+i \\textbf{w})^{*}B(\\textbf{v}+i \\textbf{w})$ \n", + "\n", + "$(\\textbf{v}+i \\textbf{w})^{*}B(\\textbf{v}+i \\textbf{w})=(\\textbf{v}+i \\textbf{w})^{*}(B(\\textbf{v}+i \\textbf{w}))=(\\textbf{v}+i \\textbf{w})^{*}(a+ib)(\\textbf{v}+i \\textbf{w})=(a+ib)(\\textbf{v}+i \\textbf{w})^{*}(\\textbf{v}+i \\textbf{w})=(a+ib)(|\\textbf{v}|^{2}+|\\textbf{w}|^{2})$ \n", + "\n", + "$(\\textbf{v}+i \\textbf{w})^{*}B(\\textbf{v}+i \\textbf{w})=((\\textbf{v}+i \\textbf{w})^{*}B^{*})(\\textbf{v}+i \\textbf{w})=(\\textbf{v}+i \\textbf{w})^{*}(a+ib)^{*}(\\textbf{v}+i \\textbf{w})=(a-ib)(\\textbf{v}+i \\textbf{w})^{*}(\\textbf{v}+i \\textbf{w})=(a-ib)(|\\textbf{v}|^{2}+|\\textbf{w}|^{2})$ \n", + "\n", + "Since $|\\textbf{v}|^{2}+|\\textbf{w}|^{2} \\neq 0$ $(a+ib) = (a-ib)$. It means that $b=0$ i.e. all eigenvalues of a real symmetric matrix are real.\n", + "\n", + "If all eigenvalues are real, at least one eigenvector is also real. Therefore $V$ is a real matrix.\n", + "\n", + "Same thing is valid for matrix $U$. Because columns of the $U$ is eigenvectors of the $AA^{*}$.\n", + "\n", + "To sum up, if $A$ is a real matrix, then it has a real SVD." + ] + }, + { + "cell_type": "code", + "execution_count": 95, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "function [U, D, V, e_A1, e_A2, norm1, norm2, norm_inf, norm_frobenius, A_inv,Area]= svdfinder (A)\n", + " dummy=0;\n", + " norm1=0;\n", + " norm2=0;\n", + " norm_inf=0;\n", + " norm_frobenius=0;\n", + " for j=1:2\n", + " for i=1:2\n", + " dummy=dummy+abs(A(i,j));\n", + " end\n", + " if dummy > norm1\n", + " norm1=dummy;\n", + " end\n", + " dummy=0;\n", + " end\n", + " for i=1:2\n", + " for j=1:2\n", + " dummy=dummy+abs(A(i,j));\n", + " end\n", + " if dummy > norm_inf\n", + " norm_inf=dummy;\n", + " end\n", + " dummy=0;\n", + " end\n", + " B=A.'*A;\n", + " \n", + " eigenA=roots([1, -(A(2,2)+A(1,1)), (A(1,1)*A(2,2)-A(2,1)*A(1,2))]);\n", + " e_A1=eigenA(1);\n", + " e_A2=eigenA(2);\n", + " eigen=roots([1, -(B(2,2)+B(1,1)), (B(1,1)*B(2,2)-B(2,1)*B(1,2))]);\n", + " e1=eigen(1);\n", + " e2=eigen(2);\n", + " D=[sqrt(eigen(1)),0; 0,sqrt(eigen(2))];\n", + " eigenvector1=[1;-((B(1,1)-e1)/B(2,1))];\n", + " eigenvector2=[1;-((B(1,1)-e2)/B(2,1))];\n", + " eigenvector1=eigenvector1/norm(eigenvector1);\n", + " eigenvector2=eigenvector2/norm(eigenvector2);\n", + " V=[eigenvector1,eigenvector2];\n", + " U=A*V;\n", + " U=[U(:,1)/D(1,1),U(:,2)/D(2,2)];\n", + " Area=pi*D(1,1)*D(2,2);\n", + " norm2=D(1,1);\n", + " \n", + " for i=1:2\n", + " norm_frobenius=norm_frobenius+(D(i,i)^2);\n", + " end\n", + " \n", + " norm_frobenius=sqrt(norm_frobenius);\n", + " \n", + " A_inv= V * [1/D(1,1),0;0,1/D(2,2)] * U.';\n", + " \n", + " \n", + " ac=[0:0.01:2*pi-0.01];\n", + " x=cos(ac);\n", + " y=sin(ac);\n", + " u1=[1;0];\n", + " u2=[0;1];\n", + " \n", + " \n", + " figure(1)\n", + " \n", + " plot(x,y,'k');\n", + " hold on\n", + " plot ([u1(1),0],[u1(2),0],'-r');\n", + " hold on\n", + " plot(u1(1),u1(2),'or');\n", + " hold on\n", + " plot ([u2(1),0],[u2(2),0]);\n", + " hold on\n", + " plot(u2(1),u2(2),'o');\n", + " hold on\n", + " plot ([eigenvector1(1),0],[eigenvector1(2),0],'-m','LineWidth',2);\n", + " hold on\n", + " plot (eigenvector1(1),eigenvector1(2),'*m');\n", + " hold on\n", + " plot ([eigenvector2(1),0],[eigenvector2(2),0],'-g','LineWidth',2);\n", + " hold on\n", + " plot (eigenvector2(1),eigenvector2(2),'*g');\n", + " \n", + " \n", + " figure(2)\n", + " \n", + " \n", + " k=U*[x;y];\n", + " k1=U*u1;\n", + " k2=U*u2;\n", + " plot(k(1,:),k(2,:),'k');\n", + " hold on\n", + " plot ([k1(1),0],[k1(2),0],'-r');\n", + " hold on\n", + " plot(k1(1),k1(2),'or');\n", + " hold on\n", + " plot ([k2(1),0],[k2(2),0]);\n", + " hold on\n", + " plot(k2(1),k2(2),'o');\n", + " e_i1=U*eigenvector1;\n", + " e_i2=U*eigenvector2;\n", + " hold on\n", + " plot ([e_i1(1),0],[e_i1(2),0],'-m','LineWidth',2);\n", + " hold on\n", + " plot (e_i1(1),e_i1(2),'*m');\n", + " hold on\n", + " plot ([e_i2(1),0],[e_i2(2),0],'-g','LineWidth',2);\n", + " hold on\n", + " plot (e_i2(1),e_i2(2),'*g');\n", + " \n", + " \n", + " figure(3)\n", + " \n", + " k=U*D*V.'*[x;y];\n", + " k1=U*D*V.'*u1;\n", + " k2=U*D*V.'*u2;\n", + " plot(k(1,:),k(2,:),'k');\n", + " hold on\n", + " plot ([k1(1),0],[k1(2),0],'-r');\n", + " hold on\n", + " plot(k1(1),k1(2),'or');\n", + " hold on\n", + " plot ([k2(1),0],[k2(2),0]);\n", + " hold on\n", + " plot(k2(1),k2(2),'o');\n", + " e_i1=U*D*V.'*eigenvector1;\n", + " e_i2=U*D*V.'*eigenvector2;\n", + " hold on\n", + " plot ([e_i1(1),0],[e_i1(2),0],'-m','LineWidth',2);\n", + " hold on\n", + " plot (e_i1(1),e_i1(2),'*m');\n", + " hold on\n", + " plot ([e_i2(1),0],[e_i2(2),0],'-g','LineWidth',2);\n", + " hold on\n", + " plot (e_i2(1),e_i2(2),'*g');\n", + "\n", + " \n", + " figure(4)\n", + " \n", + " \n", + "\n", + " k=U*D*[x;y];\n", + " k1=U*D*u1;\n", + " k2=U*D*u2;\n", + " plot(k(1,:),k(2,:),'k');\n", + " hold on\n", + " plot ([k1(1),0],[k1(2),0],'-r');\n", + " hold on\n", + " plot(k1(1),k1(2),'or');\n", + " hold on\n", + " plot ([k2(1),0],[k2(2),0]);\n", + " hold on\n", + " plot(k2(1),k2(2),'o');\n", + " e_i1=U*D*eigenvector1;\n", + " e_i2=U*D*eigenvector2;\n", + " hold on\n", + " plot ([e_i1(1),0],[e_i1(2),0],'-m','LineWidth',2);\n", + " hold on\n", + " plot (e_i1(1),e_i1(2),'*m');\n", + " hold on\n", + " plot ([e_i2(1),0],[e_i2(2),0],'-g','LineWidth',2);\n", + " hold on\n", + " plot (e_i2(1),e_i2(2),'*g');\n", + " \n", + " \n", + " figure(5)\n", + " \n", + " plot(x,y,'k');\n", + " hold on\n", + " plot ([u1(1),0],[u1(2),0],'-r');\n", + " hold on\n", + " plot(u1(1),u1(2),'or')\n", + " hold on\n", + " plot ([u2(1),0],[u2(2),0]);\n", + " hold on\n", + " plot(u2(1),u2(2),'o');\n", + " \n", + " hold on\n", + " \n", + " k=A*[x;y];\n", + " k1=A*u1;\n", + " k2=A*u2;\n", + " plot(k(1,:),k(2,:),'k');\n", + " hold on\n", + " plot ([k1(1),0],[k1(2),0],'-r');\n", + " hold on\n", + " plot(k1(1),k1(2),'or');\n", + " hold on\n", + " plot ([k2(1),0],[k2(2),0]);\n", + " hold on\n", + " plot(k2(1),k2(2),'o');\n", + " \n", + " e_i1=A*eigenvector1;\n", + " e_i2=A*eigenvector2;\n", + " hold on\n", + " plot ([e_i1(1),0],[e_i1(2),0],'-m','LineWidth',2);\n", + " hold on\n", + " plot (e_i1(1),e_i1(2),'*m');\n", + " hold on\n", + " plot ([e_i2(1),0],[e_i2(2),0],'-g','LineWidth',2);\n", + " hold on\n", + " plot (e_i2(1),e_i2(2),'*g');\n", + " \n", + " \n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 97, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "U =\n", + "\n", + " -0.70711 0.70711\n", + " -0.70711 -0.70711\n", + "\n", + "Sigma =\n", + "\n", + " 14.14214 0.00000\n", + " 0.00000 7.07107\n", + "\n", + "V =\n", + "\n", + " 0.60000 0.80000\n", + " -0.80000 0.60000\n", + "\n", + "first_eigenvalue_of_A = 1.5000 + 9.8869i\n", + "second_eigenvalue_of_A = 1.5000 - 9.8869i\n", + "norm_1 = 16\n", + "norm_2 = 14.142\n", + "norm_infinite = 15\n", + "norm_frobenius = 15.811\n", + "A_inverse =\n", + "\n", + " 0.050000 -0.110000\n", + " 0.100000 -0.020000\n", + "\n", + "Area_of_ellipsoid = 314.16\n" + ] + }, + { + "data": { + "image/png": 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2ofJ9URgGgbR0/ZCoL3MCotBP5ek22zzPGTDFjkBarrZtrbUMiRA13WbbV99wSlPUCKTF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\n/vjHf8yoaCRM2YWo7zVSlqV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RJijZ9X/vT+1nvru767+OPl0R2JLjHGI9lILQhte27XrHKY1zfCDI3d1dusNVnz9/ntUD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\nBNJqKKX6ZVNYZPTn98jL9VP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fC0WkKAqWNKZSVZVzrmmauq6ZNNrDCFN2/b/7pv3M5+fn/ddz6XAFzEI/HqJSLhJhYNq2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\n28/+q2flIyL/ZuprgogYc3H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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% This function gives the vector U, ∑ and V respectively (where A = Ux∑xV*).\n", + "% It is also plot unit vector ball (let say unit vector is \"l\"), Uxl, Ux∑xl, Ux∑xVxl and Axl with l respectively.\n", + "% Note that blue line shows vector [0;1] and red line shows vector [1;0]. (also these vectors are also multiplied with U, ∑ and V*)\n", + "% Pink and green lines show that the eigen vectors of matrix A* A. (also these vectors are also multiplied with U, ∑ and V*)\n", + "\n", + "[U, Sigma, V, first_eigenvalue_of_A, second_eigenvalue_of_A, norm_1, norm_2, norm_infinite, norm_frobenius, A_inverse,Area_of_ellipsoid]=svdfinder ([-2 11 ; -10 5])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "++++++++++++++++++++++++++++++++++++++++++++++++++++++++++$\\textbf{5.3}$\n", + "\n", + "$\\textbf{a)}$\n", + "\n", + "For matrix $A=\\begin{bmatrix} -2 & 11 \\\\ -10 & 5 \\end{bmatrix}$\n", + "\n", + "$U= \\begin{bmatrix} -0.70711 & 0.70711 \\\\ -0.70711 & -0.70711 \\end{bmatrix}$\n", + "\n", + "$\\Sigma = \\begin{bmatrix} 14.14214 & 0.00000 \\\\ 0.00000 & 7.07107 \\end{bmatrix}$\n", + "\n", + "$V = \\begin{bmatrix} 0.60000 & 0.80000 \\\\ -0.80000 & 0.60000 \\end{bmatrix}$\n", + "\n", + "Where $A=U \\Sigma V^{*}$\n", + "\n", + "$\\textbf{b)}$\n", + "\n", + "It is solved by function \"svdfinder\" above cell.\n", + "\n", + "$\\textbf{c)}$\n", + "\n", + "Since $||A||_{1}=\\underset{1 \\leq j \\leq 2}{max} \\sum_{i=1}^{2} |a_{ij}|$ i.e. max absolute column sum \n", + "\n", + "$||A||_{1}=16$\n", + "\n", + "Since $||A||_{2}=\\sigma_{1}$ where $\\sigma_{1}$ is the first element of diagonal matrix $\\Sigma$ and $A=U \\Sigma V^{*}$\n", + "\n", + "$||A||_{2} \\approx 14.142$\n", + "\n", + "Since $||A||_{\\infty}=\\underset{1 \\leq i \\leq 2}{max} \\sum_{j=1}^{2} |a_{ij}|$ i.e. max absolute row sum \n", + "\n", + "$||A||_{\\infty}=15$\n", + "\n", + "Since $||A||_{F}=\\sqrt{\\sigma_{1}^2+\\sigma_{2}^2}$ where $\\Sigma=\\begin{bmatrix} \\sigma_{1} & 0 \\\\ 0 & \\sigma_{2} \\end{bmatrix}$\n", + "\n", + "$||A||_{F} \\approx 15.811$\n", + "\n", + "These result are also found by function \"svdfinder\" \n", + "\n", + "$\\textbf{d)}$\n", + "\n", + "$A=U \\Sigma V^{*}$ and because of that matrix $U$ and $V$ are unitary, their inverses equal to their transpoze. In addition matrix $\\Sigma$ is diagonal its inverse is $\\begin{bmatrix} \\frac{1}{\\sigma_{1}} & 0 \\\\ 0 & \\frac{1}{\\sigma_{2}} \\end{bmatrix}$\n", + "\n", + "Therefore $A^{-1}=V \\Sigma^{-1} U^{*} = \\begin{bmatrix} 0.05 & -0.11 \\\\ 0.1 & -0.02 \\end{bmatrix}$\n", + "\n", + "The inverse of A is also found by function \"svdfinder\" \n", + "\n", + "$\\textbf{e)}$\n", + "\n", + "Eigen values of matrix $A$ are roots of the equation $\\lambda^{2}+\\lambda (-a-d)+ad-cb$ \n", + "\n", + "where $A=\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}$\n", + "\n", + "Hence $\\lambda_{1}=1.5000 + 9.8869i$ and $\\lambda_{2}=1.5000 - 9.8869i$\n", + "\n", + "These result are also found by function \"svdfinder\" \n", + "\n", + "$\\textbf{f)}$\n", + "\n", + "Determinant of matrix $det(A)=ad-bc=100$ and also $\\lambda_{1} \\lambda_{2} =(1.5000 + 9.8869i) \\times (1.5000 - 9.8869i) = 100$\n", + "\n", + "Absolute determinant of matrix $|det(A)|=|ad-bc|=100$ and also $\\sigma_{1} \\sigma_{2} =14.14214 \\times 7.07107 = 100$\n", + "\n", + "$\\textbf{g)}$\n", + "\n", + "The area of ellipsoid is $x*y*\\pi$ where $x$ and $y$ are ellipsoid's radiusus (pink and green lines in plotting figure). Since $x=\\sigma_{1}$ and $y=\\sigma_{2}$ The area of ellipsoid is $314.16$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "file_extension": ".m", + "help_links": [ + { + "text": "GNU Octave", + "url": "https://www.gnu.org/software/octave/support.html" + }, + { + "text": "Octave Kernel", + "url": "https://github.com/Calysto/octave_kernel" + }, + { + "text": "MetaKernel Magics", + "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" + } + ], + "mimetype": "text/x-octave", + "name": "octave", + "version": "4.0.2" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}