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cocuh/7.1.1.ipynb

Created May 31, 2015 04:40
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7.1.1.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"A = sym.Matrix([[-1/sym.sqrt(2),1,-1],[-1/sym.sqrt(2),-1,1]])"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFsAAAA0BAMAAAAXulklAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\left[\\begin{matrix}\\frac{5}{2} & - \\frac{3}{2}\\\\- \\frac{3}{2} & \\frac{5}{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡5/2 -3/2⎤\n",
"⎢ ⎥\n",
"⎣-3/2 5/2 ⎦"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A*A.T"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAYkAAAAyBAMAAABbiTqMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\left [ \\left ( 1, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 4, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}-1\\\\1\\end{matrix}\\right]\\right ]\\right )\\right ]$$"
],
"text/plain": [
"⎡⎛1, 1, ⎡⎡1⎤⎤⎞, ⎛4, 1, ⎡⎡-1⎤⎤⎞⎤\n",
"⎢⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟⎥\n",
"⎣⎝ ⎣⎣1⎦⎦⎠ ⎝ ⎣⎣1 ⎦⎦⎠⎦"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(A*A.T).eigenvects()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFsAAAA0BAMAAAAXulklAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\left[\\begin{matrix}\\frac{5}{2} & - \\frac{3}{2}\\\\- \\frac{3}{2} & \\frac{5}{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡5/2 -3/2⎤\n",
"⎢ ⎥\n",
"⎣-3/2 5/2 ⎦"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Q1=-sym.Matrix([[-1/sym.sqrt(2),1/sym.sqrt(2)],[1/sym.sqrt(2),1/sym.sqrt(2)]])\n",
"Q1*sym.Matrix([[4,0],[0,1]])*Q1.T"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAABLCAMAAABjltjdAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRO/NZondIrtsRPdGkQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAtZJREFUaAXtmuty\nozAMhc0lbDfc1+//rotNBD5ILrInZDuz5EeViCN9throCY0prH+U5lOPcQUaU9iqXh6PT4FN73Cl\ndeTiY9Ad1OeQm2roh4TVivoj+dE2+7Jiz7rlF9OMU+wwy4t6IDdlVdpzct+63kPHCJGErAfyUlkr\nyLM/Cx4K5boUWZ9Dtp78tNqTQdZnkBtbub08bb1u6exnRJ9BnuzgYMUazrjGRPRZZL/nBLKozyBH\nphfdfESfQTbrO+aR+A476nPI8+z21+vPKlGfQ16vDFXileSozyGb0V09W/2FW9QjuSpb25V99M3y\nOtAMyx85PdiIeiSfId95/Ca/c5pnve5pn03oncfvab9zmme9fsq0RUvOFz9V1TwnXLepA9ThnkVL\nTnVbbJz3HOxzSyifYB2QZUvO+pb+w0Dr/T47+E0C64AsW3LWqx1dqkz+HIh1QJYtOSN33gelk7Eu\nJEdMIiOviVHtw7AB1YXkiCXHQnr11Bp9KnjFrQ7JoiU/lL5ejpk3N7a6kJwy7covUl4UZZtu3B7d\n6/Tf6xz519fvVay38LUCTAsIY1D35yu4T6K28A8HnvQ3DQge1oXTNloLX/gd18lkqAOy0sJP4zAM\nVemvJ7QbTcQ6JIuWnDXt1htpyWSsQzKDXJi4yRcOl7W+p81GcmHinvaFw2Wtf8q0lU6f1g/OnZJC\nFHW4Z53Tp97o3CnLo6wDstLpU2907pTlUdYBWen0qTc6d8ryKOuArHT61BudO2V5lHUhOcV77v3J\nue8Z+dlRF5KTnD6135w7JSKR6ZDsnZ36Zr1nbM49QqQ004Xks2l/79wJIcfd4dPxkJx8s37595bS\n8Qs6IKudPq07dO6Uk6KkA7LW6VNzcO6UFKKoA7LS6VNvdO6U5VHWIVnn9Kk3OnfK8ijrkMyrrsvc\n5Otmyzv/v9P+V99kadwXS+o6+Q4q/90pM/6bLHVt/gJbbiqG8Gb1vgAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 2 & -2\\\\0 & -2 & 2\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 0 0 ⎤\n",
"⎢ ⎥\n",
"⎢0 2 -2⎥\n",
"⎢ ⎥\n",
"⎣0 -2 2 ⎦"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.T*A"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAocAAABMBAMAAADtigCPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+z80IA0D3mhfU6crWqWF9waD3dCf8/t/QliLx+w7b3EAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\left [ \\left ( 0, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}0\\\\1\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 1, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}1\\\\0\\\\0\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 4, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}0\\\\-1\\\\1\\end{matrix}\\right]\\right ]\\right )\\right ]$$"
],
"text/plain": [
"⎡⎛0, 1, ⎡⎡0⎤⎤⎞, ⎛1, 1, ⎡⎡1⎤⎤⎞, ⎛4, 1, ⎡⎡0 ⎤⎤⎞⎤\n",
"⎢⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟⎥\n",
"⎢⎜ ⎢⎢1⎥⎥⎟ ⎜ ⎢⎢0⎥⎥⎟ ⎜ ⎢⎢-1⎥⎥⎟⎥\n",
"⎢⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟⎥\n",
"⎣⎝ ⎣⎣1⎦⎦⎠ ⎝ ⎣⎣0⎦⎦⎠ ⎝ ⎣⎣1 ⎦⎦⎠⎦"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(A.T*A).eigenvects()"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAABLCAMAAABjltjdAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 2 & -2\\\\0 & -2 & 2\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 0 0 ⎤\n",
"⎢ ⎥\n",
"⎢0 2 -2⎥\n",
"⎢ ⎥\n",
"⎣0 -2 2 ⎦"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Q2=sym.Matrix([[0,1/sym.sqrt(2),-1/sym.sqrt(2)],[1,0,0],[0,1/sym.sqrt(2),1/sym.sqrt(2)]]).T\n",
"Q2*sym.Matrix([[4,0,0],[0,1,0],[0,0,0]])*Q2.T"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Q2*sym.Matrix([[4,0,0],[0,1,0],[0,0,0]])*Q2.T == A.T*A"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"S=sym.Matrix([[2,0,0],[0,1,0]])"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAJcAAAA/BAMAAAAf/kfJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\left[\\begin{matrix}- \\frac{\\sqrt{2}}{2} & 1 & -1\\\\- \\frac{\\sqrt{2}}{2} & -1 & 1\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ ___ ⎤\n",
"⎢-╲╱ 2 ⎥\n",
"⎢─────── 1 -1⎥\n",
"⎢ 2 ⎥\n",
"⎢ ⎥\n",
"⎢ ___ ⎥\n",
"⎢-╲╱ 2 ⎥\n",
"⎢─────── -1 1 ⎥\n",
"⎣ 2 ⎦"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Q1*S*Q2.T"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAJcAAAA/BAMAAAAf/kfJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\left[\\begin{matrix}- \\frac{\\sqrt{2}}{2} & 1 & -1\\\\- \\frac{\\sqrt{2}}{2} & -1 & 1\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ ___ ⎤\n",
"⎢-╲╱ 2 ⎥\n",
"⎢─────── 1 -1⎥\n",
"⎢ 2 ⎥\n",
"⎢ ⎥\n",
"⎢ ___ ⎥\n",
"⎢-╲╱ 2 ⎥\n",
"⎢─────── -1 1 ⎥\n",
"⎣ 2 ⎦"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.4.3"
}
},
"nbformat": 4,
"nbformat_minor": 0
}
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