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hw4 is done #9

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144 changes: 144 additions & 0 deletions .ipynb_checkpoints/Untitled-checkpoint.ipynb
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Original file line number Diff line number Diff line change
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\\bf{25.3}$\n",
"\n",
"Only matrix (a) can be obtained by left and right multiplication. Rest cannot be done.\n",
"\n",
"Matix (a) is $\\begin{bmatrix}\n",
" x & x & x \\\\ \n",
" x & x & x \\\\ \n",
" x & x & x\n",
"\\end{bmatrix}$\n",
"\n",
"\n",
"$Q_1*A=\\begin{bmatrix}\n",
" x & x & x \\\\ \n",
" 0 & x & x \\\\ \n",
" 0 & x & x\n",
"\\end{bmatrix}=A_1$\n",
"\n",
"$Q_1*A*Q_2=\\begin{bmatrix}\n",
" x & x & 0 \\\\ \n",
" 0 & x & x \\\\ \n",
" 0 & x & x\n",
"\\end{bmatrix}=A_2$\n",
"\n",
"$Q_3*Q_1*A*Q_2=\\begin{bmatrix}\n",
" x & x & 0 \\\\ \n",
" 0 & x & x \\\\ \n",
" 0 & 0 & x\n",
"\\end{bmatrix}=A_3$"
]
},
{
"cell_type": "code",
"execution_count": 141,
"metadata": {
"collapsed": false,
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A =\n",
"\n",
" 18 19 12\n",
" 10 1 12\n",
" 15 2 2\n",
"\n",
"A1 =\n",
"\n",
" 25.47548 14.99481 14.36676\n",
" 0.00000 6.35777 8.83397\n",
" 0.00000 10.03665 -2.74904\n",
"\n",
"A2 =\n",
"\n",
" 25.47548 20.76651 0.00000\n",
" 0.00000 10.70228 -1.98027\n",
" 0.00000 5.34528 8.92858\n",
"\n",
"A3 =\n",
"\n",
" 25.47548 20.76651 0.00000\n",
" 0.00000 11.96290 2.21789\n",
" 0.00000 0.00000 -8.87254\n",
"\n",
"ans =\n",
"\n",
" 25.47548 20.76651 0.00000\n",
" 0.00000 11.96290 2.21789\n",
" 0.00000 0.00000 -8.87254\n",
"\n"
]
}
],
"source": [
"A=round(rand(3,3)*20)\n",
"\n",
"e1=[1;0;0];\n",
"v1=norm(A(:,1))*e1-A(:,1);\n",
"v1=v1/norm(v1);\n",
"Q1=eye(3,3)-2*(v1*v1');\n",
"A1=Q1*A;\n",
"A1(abs(A1)<0.000000001)=0 %to get rid of error which comes from machine precision\n",
"% for example [Q , R]=qr(A) ==> Q*R-A is not zero 2\n",
"\n",
"e2=[1;0];\n",
"v2=norm(A1'(2:end,1))*e2-A1'(2:end,1);\n",
"v2=v2/norm(v2);\n",
"Q2=eye(2,2)-2*(v2*v2');\n",
"Q2=[1 0 0; 0 Q2(1,:);0 Q2(2,:)]; \n",
"Q2=Q2';\n",
"A2=A1*Q2;\n",
"A2(abs(A2)<0.000000001)=0\n",
"\n",
"\n",
"e3=[1;0];\n",
"v3=norm(A2(2:end,2))*e3-A2(2:end,2);\n",
"v3=v3/norm(v3);\n",
"Q3=eye(2,2)-2*(v3*v3');\n",
"Q3=[1 0 0; 0 Q3(1,:);0 Q3(2,:)]; \n",
"A3=Q3*A2;\n",
"A3(abs(A3)<0.000000001)=0\n",
"\n",
"Q3*Q1*A*Q2"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Octave",
"language": "octave",
"name": "octave"
},
"language_info": {
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"help_links": [
{
"text": "GNU Octave",
"url": "https://www.gnu.org/software/octave/support.html"
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{
"text": "Octave Kernel",
"url": "https://github.com/Calysto/octave_kernel"
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"text": "MetaKernel Magics",
"url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
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144 changes: 144 additions & 0 deletions Untitled.ipynb
View file
Open in desktop
Original file line number Diff line number Diff line change
@@ -0,0 +1,144 @@
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\\bf{25.3}$\n",
"\n",
"Only matrix (a) can be obtained by left and right multiplication. Rest cannot be done.\n",
"\n",
"Matix (a) is $\\begin{bmatrix}\n",
" x & x & x \\\\ \n",
" x & x & x \\\\ \n",
" x & x & x\n",
"\\end{bmatrix}$\n",
"\n",
"\n",
"$Q_1*A=\\begin{bmatrix}\n",
" x & x & x \\\\ \n",
" 0 & x & x \\\\ \n",
" 0 & x & x\n",
"\\end{bmatrix}=A_1$\n",
"\n",
"$Q_1*A*Q_2=\\begin{bmatrix}\n",
" x & x & 0 \\\\ \n",
" 0 & x & x \\\\ \n",
" 0 & x & x\n",
"\\end{bmatrix}=A_2$\n",
"\n",
"$Q_3*Q_1*A*Q_2=\\begin{bmatrix}\n",
" x & x & 0 \\\\ \n",
" 0 & x & x \\\\ \n",
" 0 & 0 & x\n",
"\\end{bmatrix}=A_3$"
]
},
{
"cell_type": "code",
"execution_count": 141,
"metadata": {
"collapsed": false,
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A =\n",
"\n",
" 18 19 12\n",
" 10 1 12\n",
" 15 2 2\n",
"\n",
"A1 =\n",
"\n",
" 25.47548 14.99481 14.36676\n",
" 0.00000 6.35777 8.83397\n",
" 0.00000 10.03665 -2.74904\n",
"\n",
"A2 =\n",
"\n",
" 25.47548 20.76651 0.00000\n",
" 0.00000 10.70228 -1.98027\n",
" 0.00000 5.34528 8.92858\n",
"\n",
"A3 =\n",
"\n",
" 25.47548 20.76651 0.00000\n",
" 0.00000 11.96290 2.21789\n",
" 0.00000 0.00000 -8.87254\n",
"\n",
"ans =\n",
"\n",
" 25.47548 20.76651 0.00000\n",
" 0.00000 11.96290 2.21789\n",
" 0.00000 0.00000 -8.87254\n",
"\n"
]
}
],
"source": [
"A=round(rand(3,3)*20)\n",
"\n",
"e1=[1;0;0];\n",
"v1=norm(A(:,1))*e1-A(:,1);\n",
"v1=v1/norm(v1);\n",
"Q1=eye(3,3)-2*(v1*v1');\n",
"A1=Q1*A;\n",
"A1(abs(A1)<0.000000001)=0 %to get rid of error which comes from machine precision\n",
"% for example [Q , R]=qr(A) ==> Q*R-A is not zero 2\n",
"\n",
"e2=[1;0];\n",
"v2=norm(A1'(2:end,1))*e2-A1'(2:end,1);\n",
"v2=v2/norm(v2);\n",
"Q2=eye(2,2)-2*(v2*v2');\n",
"Q2=[1 0 0; 0 Q2(1,:);0 Q2(2,:)]; \n",
"Q2=Q2';\n",
"A2=A1*Q2;\n",
"A2(abs(A2)<0.000000001)=0\n",
"\n",
"\n",
"e3=[1;0];\n",
"v3=norm(A2(2:end,2))*e3-A2(2:end,2);\n",
"v3=v3/norm(v3);\n",
"Q3=eye(2,2)-2*(v3*v3');\n",
"Q3=[1 0 0; 0 Q3(1,:);0 Q3(2,:)]; \n",
"A3=Q3*A2;\n",
"A3(abs(A3)<0.000000001)=0\n",
"\n",
"Q3*Q1*A*Q2"
]
}
],
"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",
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