{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Interactive eigenvectors"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"using Plots, Interact"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"plot_image (generic function with 2 methods)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function plot_image(M, x, tol=0.05)\n",
" \n",
" new_x = M * x\n",
" \n",
" unit_circle = [(cos(θ), sin(θ)) for θ in 0:0.01:2π]\n",
" ellipse = [M*[cos(θ), sin(θ)] for θ in 0:0.01:2π];\n",
"\n",
" \n",
" plot([(0, 0), (x[1], x[2])], aspect_ratio=1, lw=2, leg=false, c=:green, arrow=0.4)\n",
" plot!([(0, 0), (new_x[1], new_x[2])], lw=3, c=:blue, arrow=0.4)\n",
"\n",
" plot!(unit_circle, ls=:dash, c=:green)\n",
" plot!(first.(ellipse), last.(ellipse), ls=:dash, c=:blue)\n",
"\n",
"\n",
" # highlight when an eigenvector by changing color:\n",
" \n",
" ang1 = atan(x[2], x[1])\n",
" ang2 = atan(new_x[2], new_x[1])\n",
" \n",
" if abs(ang1 - ang2) < tol || abs(ang1 - ang2 + π) < tol || abs(ang1 - ang2 - π) < tol\n",
" plot!([(0, 0), (new_x[1], new_x[2])], lw=4, c=:magenta)\n",
" plot!([(0, 0), (x[1], x[2])], lw=4, c=:magenta)\n",
" end\n",
" \n",
" scatter!([0], [0])\n",
"\n",
" xlims!(-2, 2)\n",
" ylims!(-2, 2)\n",
"end"
]
},
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" window,\n",
" );\n",
" } else {\n",
" document\n",
" .querySelector('[data-webio-mountpoint=\"13504387864273198495\"]')\n",
" .innerHTML = '<strong>WebIO not detected.</strong>';\n",
" }\n",
" </script>\n",
"</div>\n"
],
"text/plain": [
"(div { style=Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\") }\n",
" (div { style=Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"row\") }\n",
" (div { className=\"field interact-widget\" }\n",
" (div { className=\"interact-flex-row interact-widget\" }\n",
" (div { className=\"interact-flex-row-left\" }\n",
" (label { className=\"interact \" style=Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\") }\n",
" \"a\"))\n",
" (div { className=\"interact-flex-row-center\" }\n",
" (input { max=41 min=1 attributes=Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\") step=1 className=\"slider slider is-fullwidth\" style=Dict{Any,Any}() }))\n",
" (div { className=\"interact-flex-row-right\" }\n",
" (p { attributes=Dict(\"data-bind\"=>\"text: formatted_val\") }))))\n",
" (div { className=\"field interact-widget\" }\n",
" (div { className=\"interact-flex-row interact-widget\" }\n",
" (div { className=\"interact-flex-row-left\" }\n",
" (label { className=\"interact \" style=Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\") }\n",
" \"b\"))\n",
" (div { className=\"interact-flex-row-center\" }\n",
" (input { max=41 min=1 attributes=Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\") step=1 className=\"slider slider is-fullwidth\" style=Dict{Any,Any}() }))\n",
" (div { className=\"interact-flex-row-right\" }\n",
" (p { attributes=Dict(\"data-bind\"=>\"text: formatted_val\") })))))\n",
" (div { style=Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"row\") }\n",
" (div { className=\"field interact-widget\" }\n",
" (div { className=\"interact-flex-row interact-widget\" }\n",
" (div { className=\"interact-flex-row-left\" }\n",
" (label { className=\"interact \" style=Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\") }\n",
" \"c\"))\n",
" (div { className=\"interact-flex-row-center\" }\n",
" (input { max=41 min=1 attributes=Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\") step=1 className=\"slider slider is-fullwidth\" style=Dict{Any,Any}() }))\n",
" (div { className=\"interact-flex-row-right\" }\n",
" (p { attributes=Dict(\"data-bind\"=>\"text: formatted_val\") }))))\n",
" (div { className=\"field interact-widget\" }\n",
" (div { className=\"interact-flex-row interact-widget\" }\n",
" (div { className=\"interact-flex-row-left\" }\n",
" (label { className=\"interact \" style=Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\") }\n",
" \"d\"))\n",
" (div { className=\"interact-flex-row-center\" }\n",
" (input { max=41 min=1 attributes=Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\") step=1 className=\"slider slider is-fullwidth\" style=Dict{Any,Any}() }))\n",
" (div { className=\"interact-flex-row-right\" }\n",
" (p { attributes=Dict(\"data-bind\"=>\"text: formatted_val\") }))))))"
]
},
"execution_count": 4,
"metadata": {
"application/vnd.webio.node+json": {
"kernelId": "b60d3081-e7dd-4d68-bb83-50e4fad3e468"
}
},
"output_type": "execute_result"
}
],
"source": [
"a = slider(-2:0.1:2, label=\"a\")\n",
"b = slider(-2:0.1:2, label=\"b\")\n",
"c = slider(-2:0.1:2, label=\"c\")\n",
"d = slider(-2:0.1:2, label=\"d\")\n",
"\n",
"\n",
"sliders = vbox( hbox(a, b),\n",
" hbox(c, d)\n",
" )"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
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{
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1342.74,427.59 1339.26,426.529 1335.77,425.503 \\n 1332.27,424.511 1328.75,423.555 1325.23,422.634 1321.7,421.749 1318.16,420.898 1314.62,420.083 1311.06,419.304 1307.5,418.56 1303.93,417.852 1300.35,417.179 \\n 1296.76,416.543 1293.17,415.942 1289.58,415.377 1285.98,414.848 1282.37,414.356 1278.76,413.899 1275.14,413.478 1271.52,413.094 1267.9,412.746 1264.27,412.434 \\n 1260.64,412.158 1257.01,411.919 1253.38,411.716 1249.74,411.549 1246.1,411.419 1242.47,411.325 1238.83,411.268 1235.19,411.247 1231.55,411.262 1227.91,411.314 \\n 1224.27,411.402 1220.63,411.526 1216.99,411.687 1213.36,411.884 1209.73,412.118 1206.1,412.388 1202.47,412.694 1198.85,413.036 1195.22,413.415 1191.61,413.83 \\n 1188,414.281 1184.39,414.768 1180.79,415.291 1177.19,415.85 1173.6,416.445 1170.01,417.076 1166.44,417.742 1162.86,418.445 1159.3,419.183 1155.74,419.957 \\n 1152.19,420.766 1148.65,421.611 1145.12,422.491 1141.6,423.406 1138.08,424.357 1134.58,425.342 1131.09,426.363 1127.6,427.419 1124.13,428.509 1120.67,429.634 \\n 1117.22,430.793 1113.78,431.987 1110.35,433.216 1106.94,434.478 1103.54,435.775 1100.15,437.105 1096.77,438.47 1093.41,439.868 1090.07,441.299 1086.74,442.764 \\n 1083.42,444.262 1080.12,445.794 1076.83,447.358 1073.56,448.955 1070.3,450.585 1067.06,452.247 1063.84,453.941 1060.64,455.668 1057.45,457.426 1054.28,459.217 \\n 1051.13,461.039 1048,462.892 1044.88,464.777 1041.79,466.692 1038.71,468.639 1035.66,470.616 1032.62,472.623 1029.6,474.661 1026.61,476.729 1023.63,478.827 \\n 1020.68,480.954 1017.75,483.111 1014.84,485.297 1011.95,487.512 1009.08,489.756 1006.24,492.028 1003.42,494.329 1000.62,496.658 997.847,499.015 995.097,501.399 \\n 992.37,503.81 989.668,506.249 986.99,508.715 984.337,511.207 981.709,513.726 979.107,516.271 976.53,518.842 973.979,521.438 971.454,524.06 968.955,526.707 \\n 966.483,529.379 964.038,532.075 961.62,534.796 959.229,537.541 956.866,540.309 954.53,543.101 952.223,545.916 949.944,548.754 947.693,551.615 945.471,554.498 \\n 943.278,557.404 941.114,560.331 938.98,563.279 936.875,566.249 934.8,569.24 932.755,572.251 930.74,575.282 928.755,578.334 926.802,581.405 924.879,584.496 \\n 922.987,587.605 921.126,590.734 919.296,593.881 917.499,597.046 915.732,600.228 913.998,603.429 912.296,606.646 910.626,609.881 908.989,613.131 907.384,616.399 \\n 905.812,619.682 904.273,622.98 902.766,626.294 901.294,629.623 899.854,632.966 898.448,636.323 897.076,639.695 895.737,643.08 894.432,646.478 893.162,649.889 \\n 891.925,653.312 890.723,656.748 889.555,660.196 888.422,663.655 887.323,667.125 886.259,670.606 885.23,674.098 884.236,677.599 883.277,681.111 882.354,684.632 \\n 881.465,688.162 880.612,691.7 879.794,695.247 879.012,698.802 878.265,702.365 877.554,705.935 876.879,709.511 876.24,713.095 875.636,716.684 875.068,720.28 \\n 874.537,723.881 874.041,727.487 873.581,731.098 873.158,734.713 872.771,738.333 872.42,741.956 872.105,745.582 871.826,749.211 871.584,752.843 871.378,756.477 \\n 871.209,760.113 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1324.11,1128.15 1327.63,1127.24 1331.15,1126.29 1334.65,1125.31 \\n 1338.15,1124.3 1341.64,1123.25 1345.11,1122.16 1348.57,1121.05 1352.03,1119.89 1355.47,1118.7 1358.89,1117.48 1362.31,1116.22 1365.71,1114.93 1369.1,1113.61 \\n 1372.48,1112.25 1375.84,1110.86 1379.19,1109.43 1382.53,1107.97 1385.85,1106.48 1389.15,1104.95 1392.44,1103.39 1395.71,1101.8 1398.97,1100.17 1402.21,1098.52 \\n 1405.44,1096.83 1408.64,1095.11 1411.83,1093.35 1415.01,1091.57 1418.16,1089.75 1421.3,1087.9 1424.41,1086.02 1427.51,1084.11 1430.59,1082.17 1433.65,1080.2 \\n 1436.69,1078.2 1439.71,1076.16 1442.71,1074.1 1445.69,1072.01 1448.64,1069.88 1451.58,1067.73 1454.49,1065.55 1457.38,1063.34 1460.25,1061.1 1463.1,1058.83 \\n 1465.93,1056.54 1468.73,1054.21 1471.51,1051.86 1474.26,1049.48 1476.99,1047.07 1479.7,1044.64 1482.38,1042.18 1485.03,1039.69 1487.67,1037.17 1490.27,1034.63 \\n 1492.85,1032.07 1495.41,1029.47 1497.94,1026.86 1500.44,1024.21 1502.92,1021.55 1505.37,1018.85 1507.79,1016.14 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" window,\n",
" );\n",
" } else {\n",
" document\n",
" .querySelector('[data-webio-mountpoint=\"13464725398146600755\"]')\n",
" .innerHTML = '<strong>WebIO not detected.</strong>';\n",
" }\n",
" </script>\n",
"</div>\n"
],
"text/plain": [
"Observable{Any} with 0 listeners. Value:\n",
"Plot{Plots.GRBackend() n=7}"
]
},
"execution_count": 5,
"metadata": {
"application/vnd.webio.node+json": {
"kernelId": "b60d3081-e7dd-4d68-bb83-50e4fad3e468"
}
},
"output_type": "execute_result"
}
],
"source": [
"r = -2:0.01:2\n",
"\n",
"θ_vals = range(-2π, 2π, length = 500)\n",
"formatted_vals = map(t -> \"$(floor(t/π, digits=2))π\", θ_vals)\n",
"\n",
"#@manipulate for a in r, b in r, c in r, d in r, θ in \n",
"\n",
"θ = slider(θ_vals, formatted_vals, label=\"θ\")\n",
"\n",
"p = map(a, b, c, d, θ) do a, b, c, d, θ\n",
"\n",
" M = [a b; c d]\n",
" \n",
" x = [cos(θ), sin(θ)]\n",
" \n",
" plot_image(M, x)\n",
"\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
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"children": [
{
"children": [
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{
"children": [
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"children": [
{
"children": [
"a"
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"instanceArgs": {
"namespace": "html",
"tag": "label"
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"nodeType": "DOM",
"props": {
"className": "interact ",
"style": {
"padding": "5px 10px 0px 10px"
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},
"type": "node"
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],
"instanceArgs": {
"namespace": "html",
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"props": {
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{
"children": [
{
"children": [],
"instanceArgs": {
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"tag": "input"
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"nodeType": "DOM",
"props": {
"attributes": {
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"orient": "horizontal",
"type": "range"
},
"className": "slider slider is-fullwidth",
"max": 41,
"min": 1,
"step": 1,
"style": {}
},
"type": "node"
}
],
"instanceArgs": {
"namespace": "html",
"tag": "div"
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"nodeType": "DOM",
"props": {
"className": "interact-flex-row-center"
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"type": "node"
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{
"children": [
{
"children": [],
"instanceArgs": {
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"tag": "p"
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"props": {
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"type": "node"
}
],
"instanceArgs": {
"namespace": "html",
"tag": "div"
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"nodeType": "DOM",
"props": {
"className": "interact-flex-row-right"
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"type": "node"
}
],
"instanceArgs": {
"namespace": "html",
"tag": "div"
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"nodeType": "DOM",
"props": {
"className": "interact-flex-row interact-widget"
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"type": "node"
}
],
"instanceArgs": {
"handlers": {
"changes": [
"(function (val){return (val!=this.model[\"changes\"]()) ? (this.valueFromJulia[\"changes\"]=true, this.model[\"changes\"](val)) : undefined})"
],
"index": [
"(function (val){return (val!=this.model[\"index\"]()) ? (this.valueFromJulia[\"index\"]=true, this.model[\"index\"](val)) : undefined})"
]
},
"id": "knockout-component-24c8dad5-b400-4069-baab-4e449bc928ec",
"imports": {
"data": [
{
"name": "knockout",
"type": "js",
"url": "/assetserver/f07383ad7ba5d9e293900fdc7e5e611110d6ff32-knockout.js"
},
{
"name": "knockout_punches",
"type": "js",
"url": "/assetserver/306cc87ce82e45c3e5eb81524cb970ae2bab3932-knockout_punches.js"
},
{
"name": null,
"type": "js",
"url": "/assetserver/69078235d89420aa60ec07ae908cfcc65b6ecf85-all.js"
},
{
"name": null,
"type": "css",
"url": "/assetserver/98bc206151bf33944d9780fd349511e9eae784a0-style.css"
},
{
"name": null,
"type": "css",
"url": "/assetserver/13615fa9690b1c4388c3fe14db802e442333ebc4-bulma_confined.min.css"
}
],
"type": "async_block"
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},
"execution_count": 9,
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"source": [
"vbox(sliders, θ, p)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Define matrix using the SVD"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
"using LinearAlgebra"
]
},
{
"cell_type": "code",
"execution_count": 11,
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(WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-86b94cfa-812e-48a9-9b51-d6d0694f4cff\\\",\\\"id\\\":\\\"ob_44\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"knockout-component-86b94cfa-812e-48a9-9b51-d6d0694f4cff\\\",\\\"id\\\":\\\"ob_43\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"index\\\"]=false}),self)]\\n \\n }\\n self.model = new AppViewModel();\\n self.valueFromJulia = {};\\n for (var key in json_data) {\\n self.valueFromJulia[key] = false;\\n }\\n ko.applyBindings(self.model, self.dom);\\n}\\n);\\n (WebIO.importBlock({\\\"data\\\":[{\\\"name\\\":\\\"knockout\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"\\/assetserver\\/f07383ad7ba5d9e293900fdc7e5e611110d6ff32-knockout.js\\\"},{\\\"name\\\":\\\"knockout_punches\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"\\/assetserver\\/306cc87ce82e45c3e5eb81524cb970ae2bab3932-knockout_punches.js\\\"}],\\\"type\\\":\\\"async_block\\\"})).then((imports) => handler.apply(this, imports));\\n}\\n\"],\"observables\":{\"changes\":{\"sync\":false,\"id\":\"ob_44\",\"value\":0},\"index\":{\"sync\":true,\"id\":\"ob_43\",\"value\":101}}},\"children\":[{\"props\":{\"className\":\"interact-flex-row interact-widget\"},\"nodeType\":\"DOM\",\"type\":\"node\",\"instanceArgs\":{\"namespace\":\"html\",\"tag\":\"div\"},\"children\":[{\"props\":{\"className\":\"interact-flex-row-left\"},\"nodeType\":\"DOM\",\"type\":\"node\",\"instanceArgs\":{\"namespace\":\"html\",\"tag\":\"div\"},\"children\":[{\"props\":{\"className\":\"interact \",\"style\":{\"padding\":\"5px 10px 0px 10px\"}},\"nodeType\":\"DOM\",\"type\":\"node\",\"instanceArgs\":{\"namespace\":\"html\",\"tag\":\"label\"},\"children\":[\"σ₁\"]}]},{\"props\":{\"className\":\"interact-flex-row-center\"},\"nodeType\":\"DOM\",\"type\":\"node\",\"instanceArgs\":{\"namespace\":\"html\",\"tag\":\"div\"},\"children\":[{\"props\":{\"max\":201,\"min\":1,\"attributes\":{\"type\":\"range\",\"data-bind\":\"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\":\"horizontal\"},\"step\":1,\"className\":\"slider slider is-fullwidth\",\"style\":{}},\"nodeType\":\"DOM\",\"type\":\"node\",\"instanceArgs\":{\"namespace\":\"html\",\"tag\":\"input\"},\"children\":[]}]},{\"props\":{\"className\":\"interact-flex-row-right\"},\"nodeType\":\"DOM\",\"type\":\"node\",\"instanceArgs\":{\"namespace\":\"html\",\"tag\":\"div\"},\"children\":[{\"props\":{\"attributes\":{\"data-bind\":\"text: formatted_val\"}},\"nodeType\":\"DOM\",\"type\":\"node\",\"instanceArgs\":{\"namespace\":\"html\",\"tag\":\"p\"},\"children\":[]}]}]}]}]},{\"props\":{\"className\":\"field interact-widget\"},\"nodeType\":\"DOM\",\"type\":\"node\",\"instanceArgs\":{\"namespace\":\"html\",\"tag\":\"div\"},\"children\":[{\"props\":{},\"nodeType\":\"Scope\",\"type\":\"node\",\"instanceArgs\":{\"imports\":{\"data\":[{\"name\":\"knockout\",\"type\":\"js\",\"url\":\"\\/assetserver\\/f07383ad7ba5d9e293900fdc7e5e611110d6ff32-knockout.js\"},{\"name\":\"knockout_punches\",\"type\":\"js\",\"url\":\"\\/assetserver\\/306cc87ce82e45c3e5eb81524cb970ae2bab3932-knockout_punches.js\"},{\"name\":null,\"type\":\"js\",\"url\":\"\\/assetserver\\/69078235d89420aa60ec07ae908cfcc65b6ecf85-all.js\"},{\"name\":null,\"type\":\"css\",\"url\":\"\\/assetserver\\/98bc206151bf33944d9780fd349511e9eae784a0-style.css\"},{\"name\":null,\"type\":\"css\",\"url\":\"\\/assetserver\\/13615fa9690b1c4388c3fe14db802e442333ebc4-bulma_confined.min.css\"}],\"type\":\"async_block\"},\"id\":\"knockout-component-44316dd1-1d7a-412a-bbca-b1775f2edf06\",\"handlers\":{\"changes\":[\"(function (val){return (val!=this.model[\\\"changes\\\"]()) ? 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(this.valueFromJulia[\\\"index\\\"]=true, this.model[\\\"index\\\"](val)) : undefined})\"]},\"systemjs_options\":null,\"mount_callbacks\":[\"function () {\\n var handler = (function (ko, koPunches) {\\n ko.punches.enableAll();\\n ko.bindingHandlers.numericValue = {\\n init : function(element, valueAccessor, allBindings, data, context) {\\n var stringified = ko.observable(ko.unwrap(valueAccessor()));\\n stringified.subscribe(function(value) {\\n var val = parseFloat(value);\\n if (!isNaN(val)) {\\n valueAccessor()(val);\\n }\\n })\\n valueAccessor().subscribe(function(value) {\\n var str = JSON.stringify(value);\\n if ((str == \\\"0\\\") && ([\\\"-0\\\", \\\"-0.\\\"].indexOf(stringified()) >= 0))\\n return;\\n if ([\\\"null\\\", \\\"\\\"].indexOf(str) >= 0)\\n return;\\n stringified(str);\\n })\\n ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get('valueUpdate')}, context);\\n }\\n };\\n var json_data = {\\\"formatted_vals\\\":[\\\"0.0\\\",\\\"0.01\\\",\\\"0.02\\\",\\\"0.03\\\",\\\"0.04\\\",\\\"0.05\\\",\\\"0.06\\\",\\\"0.07\\\",\\\"0.08\\\",\\\"0.09\\\",\\\"0.1\\\",\\\"0.11\\\",\\\"0.12\\\",\\\"0.13\\\",\\\"0.14\\\",\\\"0.15\\\",\\\"0.16\\\",\\\"0.17\\\",\\\"0.18\\\",\\\"0.19\\\",\\\"0.2\\\",\\\"0.21\\\",\\\"0.22\\\",\\\"0.23\\\",\\\"0.24\\\",\\\"0.25\\\",\\\"0.26\\\",\\\"0.27\\\",\\\"0.28\\\",\\\"0.29\\\",\\\"0.3\\\",\\\"0.31\\\",\\\"0.32\\\",\\\"0.33\\\",\\\"0.34\\\",\\\"0.35\\\",\\\"0.36\\\",\\\"0.37\\\",\\\"0.38\\\",\\\"0.39\\\",\\\"0.4\\\",\\\"0.41\\\",\\\"0.42\\\",\\\"0.43\\\",\\\"0.44\\\",\\\"0.45\\\",\\\"0.46\\\",\\\"0.47\\\",\\\"0.48\\\",\\\"0.49\\\",\\\"0.5\\\",\\\"0.51\\\",\\\"0.52\\\",\\\"0.53\\\",\\\"0.54\\\",\\\"0.55\\\",\\\"0.56\\\",\\\"0.57\\\",\\\"0.58\\\",\\\"0.59\\\",\\\"0.6\\\",\\\"0.61\\\",\\\"0.62\\\",\\\"0.63\\\",\\\"0.64\\\",\\\"0.65\\\",\\\"0.66\\\",\\\"0.67\\\",\\\"0.68\\\",\\\"0.69\\\",\\\"0.7\\\",\\\"0.71\\\",\\\"0.72\\\",\\\"0.73\\\",\\\"0.74\\\",\\\"0.75\\\",\\\"0.76\\\",\\\"0.77\\\",\\\"0.78\\\",\\\"0.79\\\",\\\"0.8\\\",\\\"0.81\\\",\\\"0.82\\\",\\\"0.83\\\",\\\"0.84\\\",\\\"0.85\\\",\\\"0.86\\\",\\\"0.87\\\",\\\"0.88\\\",\\\"0.89\\\",\\\"0.9\\\",\\\"0.91\\\",\\\"0.92\\\",\\\"0.93\\\",\\\"0.94\\\",\\\"0.95\\\",\\\"0.96\\\",\\\"0.97\\\",\\\"0.98\\\",\\\"0.99\\\",\\\"1.0\\\",\\\"1.01\\\",\\\"1.02\\\",\\\"1.03\\\",\\\"1.04\\\",\\\"1.05\\\",\\\"1.06\\\",\\\"1.07\\\",\\\"1.08\\\",\\\"1.09\\\",\\\"1.1\\\",\\\"1.11\\\",\\\"1.12\\\",\\\"1.13\\\",\\\"1.14\\\",\\\"1.15\\\",\\\"1.16\\\",\\\"1.17\\\",\\\"1.18\\\",\\\"1.19\\\",\\\"1.2\\\",\\\"1.21\\\",\\\"1.22\\\",\\\"1.23\\\",\\\"1.24\\\",\\\"1.25\\\",\\\"1.26\\\",\\\"1.27\\\",\\\"1.28\\\",\\\"1.29\\\",\\\"1.3\\\",\\\"1.31\\\",\\\"1.32\\\",\\\"1.33\\\",\\\"1.34\\\",\\\"1.35\\\",\\\"1.36\\\",\\\"1.37\\\",\\\"1.38\\\",\\\"1.39\\\",\\\"1.4\\\",\\\"1.41\\\",\\\"1.42\\\",\\\"1.43\\\",\\\"1.44\\\",\\\"1.45\\\",\\\"1.46\\\",\\\"1.47\\\",\\\"1.48\\\",\\\"1.49\\\",\\\"1.5\\\",\\\"1.51\\\",\\\"1.52\\\",\\\"1.53\\\",\\\"1.54\\\",\\\"1.55\\\",\\\"1.56\\\",\\\"1.57\\\",\\\"1.58\\\",\\\"1.59\\\",\\\"1.6\\\",\\\"1.61\\\",\\\"1.62\\\",\\\"1.63\\\",\\\"1.64\\\",\\\"1.65\\\",\\\"1.66\\\",\\\"1.67\\\",\\\"1.68\\\",\\\"1.69\\\",\\\"1.7\\\",\\\"1.71\\\",\\\"1.72\\\",\\\"1.73\\\",\\\"1.74\\\",\\\"1.75\\\",\\\"1.76\\\",\\\"1.77\\\",\\\"1.78\\\",\\\"1.79\\\",\\\"1.8\\\",\\\"1.81\\\",\\\"1.82\\\",\\\"1.83\\\",\\\"1.84\\\",\\\"1.85\\\",\\\"1.86\\\",\\\"1.87\\\",\\\"1.88\\\",\\\"1.89\\\",\\\"1.9\\\",\\\"1.91\\\",\\\"1.92\\\",\\\"1.93\\\",\\\"1.94\\\",\\\"1.95\\\",\\\"1.96\\\",\\\"1.97\\\",\\\"1.98\\\",\\\"1.99\\\",\\\"2.0\\\"],\\\"changes\\\":WebIO.getval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-44316dd1-1d7a-412a-bbca-b1775f2edf06\\\",\\\"id\\\":\\\"ob_47\\\",\\\"type\\\":\\\"observable\\\"}),\\\"index\\\":WebIO.getval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"knockout-component-44316dd1-1d7a-412a-bbca-b1775f2edf06\\\",\\\"id\\\":\\\"ob_46\\\",\\\"type\\\":\\\"observable\\\"})};\\n var self = this;\\n function AppViewModel() {\\n for (var key in json_data) {\\n var el = json_data[key];\\n this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n }\\n \\n [this[\\\"formatted_val\\\"]=ko.computed( function(){\\n return this.formatted_vals()[parseInt(this.index())-1];\\n }\\n,this)]\\n [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-44316dd1-1d7a-412a-bbca-b1775f2edf06\\\",\\\"id\\\":\\\"ob_47\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? 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var self = this;\\n function AppViewModel() {\\n for (var key in json_data) {\\n var el = json_data[key];\\n this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n }\\n \\n [this[\\\"formatted_val\\\"]=ko.computed( function(){\\n return this.formatted_vals()[parseInt(this.index())-1];\\n }\\n,this)]\\n [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-c37a04b4-0615-4aee-88ba-6db78ee26adb\\\",\\\"id\\\":\\\"ob_56\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"knockout-component-c37a04b4-0615-4aee-88ba-6db78ee26adb\\\",\\\"id\\\":\\\"ob_55\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"index\\\"]=false}),self)]\\n \\n }\\n self.model = new AppViewModel();\\n self.valueFromJulia = {};\\n for (var key in json_data) {\\n self.valueFromJulia[key] = false;\\n }\\n ko.applyBindings(self.model, self.dom);\\n}\\n);\\n (WebIO.importBlock({\\\"data\\\":[{\\\"name\\\":\\\"knockout\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/f07383ad7ba5d9e293900fdc7e5e611110d6ff32-knockout.js\\\"},{\\\"name\\\":\\\"knockout_punches\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/306cc87ce82e45c3e5eb81524cb970ae2bab3932-knockout_punches.js\\\"}],\\\"type\\\":\\\"async_block\\\"})).then((imports) => handler.apply(this, imports));\\n}\\n\")])], Dict{Symbol,Any}(:className=>\"field interact-widget\"), 1), Observable{Any} with 0 listeners. Value:\n",
"Node{DOM}(DOM(:html, :div), Any[Plot{Plots.GRBackend() n=7}], Dict{Symbol,Any}(:className=>\"interact-flex-row interact-widget\"), 1)], Dict{Symbol,Any}(), 6)"
]
},
"execution_count": 11,
"metadata": {
"application/vnd.webio.node+json": {
"kernelId": "b60d3081-e7dd-4d68-bb83-50e4fad3e468"
}
},
"output_type": "execute_result"
}
],
"source": [
"@manipulate for σ₁ in 0:0.01:2, σ₂ in 0:0.01:2, θ in 0:0.01:2π, ϕ in 0:0.01:2π, angle=-2π:0.01:2π\n",
"\n",
" # when phi=theta have U = V so a symmetric matrix\n",
" \n",
" U = [cos(θ) sin(θ); -sin(θ) cos(θ)]\n",
" V = [cos(ϕ) sin(ϕ); -sin(ϕ) cos(ϕ)]\n",
"\n",
" Σ = Diagonal([σ₁, σ₂])\n",
"\n",
" M = U * Σ * V'\n",
" \n",
" x = [cos(angle), sin(angle)]\n",
"\n",
" plot_image(M, x, 0.01)\n",
" \n",
" title!(\"evals = $(eigvals(M))\")\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
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"display_name": "Julia 1.1.0",
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