lec05.ipynb

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   "id": "25289497",
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/srv/rep/22-course-cse6643/mybinder/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/srv/rep/22-course-cse6643/mybinder/Manifest.toml`\n"
     ]
    }
   ],
   "source": [
    "import Pkg; Pkg.add([\"Plots\", \"Interact\"]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "9df477db",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "const STATIC_LECTURE = false;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "5541fc3c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "draw (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function draw(plt)\n",
    "    if STATIC_LECTURE\n",
    "        filename = Base.Filesystem.tempname() * \".png\"\n",
    "        savefig(filename)\n",
    "        return display(\"image/png\", read(filename))\n",
    "    else\n",
    "        return plt\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e4363cb2",
   "metadata": {},
   "outputs": [
    {
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       "<div style=\"padding: 1em; background-color: #f8d6da; border: 1px solid #f5c6cb; font-weight: bold;\">\n",
       "<p>The WebIO Jupyter extension was not detected. See the\n",
       "<a href=\"https://juliagizmos.github.io/WebIO.jl/latest/providers/ijulia/\" target=\"_blank\">\n",
       "    WebIO Jupyter integration documentation\n",
       "</a>\n",
       "for more information.\n",
       "</div>\n"
      ],
      "text/plain": [
       "WebIO._IJuliaInit()"
      ]
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    {
     "data": {
      "text/plain": [
       "imshow (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra;\n",
    "using Plots; default(l=2, size=(900,550), label=false);\n",
    "using Interact;\n",
    "\n",
    "imshow(x) = heatmap(x\n",
    "    , aspect_ratio=:equal\n",
    "    , framestyle=:none\n",
    "    , yflip=true\n",
    "    , c=:diverging_gwv_55_95_c39_n256\n",
    "    , clim=if maximum(abs.(x)) > 1.e-12 (-maximum(abs.(x)),maximum(abs.(x))) else (-1,1) end\n",
    "    , size=(500,500)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1d7f2b37",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**(<https://tinyurl.com/gtcse6643lec5>; more install details: <https://www.juliaopt.org/install.pdf>)**\n",
    "\n",
    "### Warmup\n",
    "\n",
    "1. If you have not already, install julia: <https://julialang.org/downloads>\n",
    "\n",
    "2. Run `julia` to open the julia [REPL](https://en.wikipedia.org/wiki/Read%E2%80%93eval%E2%80%93print_loop)\n",
    "\n",
    "```\n",
    "   _       _ _(_)_     |  Documentation: https://docs.julialang.org\n",
    "  (_)     | (_) (_)    |\n",
    "   _ _   _| |_  __ _   |  Type \"?\" for help, \"]?\" for Pkg help.\n",
    "  | | | | | | |/ _` |  |\n",
    "  | | |_| | | | (_| |  |  Version 1.7.0 (2021-11-30)\n",
    " _/ |\\__'_|_|_|\\__'_|  |  \n",
    "|__/                   |\n",
    "\n",
    "julia> \n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea32988b",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "3. In julia, you can install [IJulia](https://juliapackages.com/p/ijulia):\n",
    "\n",
    "```julia\n",
    "julia> import Pkg;\n",
    "julia> Pkg.add(\"IJulia\")\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "937606a8",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "4. Test it:\n",
    "\n",
    "```julia\n",
    "julia> using IJulia\n",
    "julia> notebook()\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31d860cb",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Projectors and the QR Factorization\n",
    "\n",
    "**1. Projectors and Complementary Projectors**\n",
    "\n",
    "**2. Orthogonal Projectors**\n",
    "\n",
    "**3. Projection with an Orthogonal vs. Arbitrary Basis**\n",
    "\n",
    "**4. QR Factorization: Introduction**\n",
    "\n",
    "**5. QR by Projections**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dcf2def3",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Projectors"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c614b6a",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "A _projector_ is a matrix $P$ that satisfies\n",
    "\n",
    "$$P^2 = P.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36097942",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## $P^2 = P$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd15f4c1",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $P$ must be square"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee17aa6b",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- As an operator, _idempotent_:\n",
    "\n",
    "$$P^2 x = Px$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ad1d613",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- If $(\\lambda, v)$ is an eigenpair?\n",
    "\n",
    "$$ P^2 v = P v \\quad \\Leftrightarrow \\quad \\lambda^2 v = \\lambda v\n",
    "\\quad \\Leftrightarrow \\quad \\lambda(\\lambda - 1) = 0 \n",
    "\\quad \\Leftrightarrow \\quad \\lambda \\in \\{0,1\\}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c4086bc",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $\\lambda = 1$? If and only if $v \\in \\mathrm{range}(P)$\n",
    "\n",
    "$$ P^2 v = P v  \\quad \\Leftrightarrow \\quad P (\\lambda v) = \\lambda v\n",
    "\\quad \\Leftrightarrow \\quad P v = v $$"
   ]
  },
  {
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   "source": [
    "### Intuition: \"projected\" like a shadow on a wall\n",
    "\n",
    "![projector.svg](attachment:projector.svg)"
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   "cell_type": "markdown",
   "id": "f5f21724",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Shifting the shadow to the origin\n",
    "\n",
    "Shows $(I - P)v \\in \\mathrm{null}(P)$ for every $v$\n",
    "\n",
    "![projector2.svg](attachment:projector2.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fda89417",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "(Algebraically: $P(I - P)v = Pv - P^2 v = Pv - Pv = 0$.)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "316ac25b",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   },
   "source": [
    "### Complementary Projectors\n",
    "\n",
    "If $P$ is a projector, $I - P$ is also a projector, the _complementary projector_:\n",
    "\n",
    "$$(I - P)^2 = I - 2 P + P^2 = I - 2 P + P = I - P.$$"
   ]
  },
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   },
   "cell_type": "markdown",
   "id": "ebd51bb8",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The range of one is the nullspace of the other\n",
    "\n",
    "$$\\mathrm{null}(P) = \\mathrm{range}(I - P), \\quad \\mathrm{null}(I-P) = \\mathrm{range}(P)$$\n",
    "\n",
    "![projector3.svg](attachment:projector3.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "362eab4e",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "> A projector separates $\\mathbb{C}^m$ into two spaces."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "bdb983b0",
   "metadata": {
    "hideCode": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "function plot_projections(x, y, θ, ϕ)\n",
    "    scale = 1.1 * max(abs(x), abs(y))\n",
    "    u = [cos(θ), sin(θ)]\n",
    "    v = [-sin(ϕ), cos(ϕ)]\n",
    "    d = v'*u\n",
    "    scale = 1.1\n",
    "    p = plot(ylim = (-scale, scale)\n",
    "        , xlim = (-1.618*scale, 1.618*scale)\n",
    "        , framestyle = :origin\n",
    "        , aspect_ratio = :equal\n",
    "    )\n",
    "    plot!(p, [0,x], [0,y], l = :green, text = [\"\", \"x\"])\n",
    "    if (abs(d) > 100 * eps(Float64))\n",
    "        P = u*v' / d\n",
    "        Px, Py = P * [x, y]\n",
    "        plot!(p, [0, Px], [0, Py]\n",
    "            , color = :blue\n",
    "            , text = [\"\", \"Px\"]\n",
    "            )\n",
    "        ImPx, ImPy = (I - P) * [x, y]\n",
    "        plot!(p, [0, ImPx], [0, ImPy]\n",
    "            , color = :orange\n",
    "            , text = [\"\", \"(I-P)x\"]\n",
    "            )\n",
    "        plot!(p, [ImPx, x], [ImPy, y]\n",
    "            , l = (1, :blue, :dot)\n",
    "            )\n",
    "        plot!(p, [Px, x], [Py, y]\n",
    "            , l = (1, :orange, :dot)\n",
    "            )\n",
    "    end\n",
    "    function plot_axis!(plt, angle)\n",
    "        c = cos(angle)\n",
    "        s = sin(angle)\n",
    "        m = max(abs(x), abs(y))\n",
    "        c *= scale / m\n",
    "        s *= scale / m\n",
    "        plot!(plt, [-c,c], [-s,s]\n",
    "            , line = (1, :dash, :black)\n",
    "        )\n",
    "        return [[-x,x], [-y,y]]\n",
    "    end\n",
    "    plot_axis!(p, θ)\n",
    "    plot_axis!(p, ϕ)\n",
    "    return p\n",
    "end;"
   ]
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        "tag": "div"
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       "<div\n",
       "    class=\"webio-mountpoint\"\n",
       "    data-webio-mountpoint=\"7093032502373938503\"\n",
       ">\n",
       "    <script>\n",
       "    (function(){\n",
       "    // Some integrations (namely, IJulia/Jupyter) use an alternate render pathway than\n",
       "    // just putting the html on the page. If WebIO isn't defined, then it's pretty likely\n",
       "    // that we're in one of those situations and the integration just isn't installed\n",
       "    // correctly.\n",
       "    if (typeof window.WebIO === \"undefined\") {\n",
       "        document\n",
       "            .querySelector('[data-webio-mountpoint=\"7093032502373938503\"]')\n",
       "            .innerHTML = (\n",
       "                '<div style=\"padding: 1em; background-color: #f8d6da; border: 1px solid #f5c6cb; font-weight: bold;\">' +\n",
       "                '<p><strong>WebIO not detected.</strong></p>' +\n",
       "                '<p>Please read ' +\n",
       "                '<a href=\"https://juliagizmos.github.io/WebIO.jl/latest/troubleshooting/not-detected/\" target=\"_blank\">the troubleshooting guide</a> ' +\n",
       "                'for more information on how to resolve this issue.</p>' +\n",
       "                '<p><a href=\"https://juliagizmos.github.io/WebIO.jl/latest/troubleshooting/not-detected/\" target=\"_blank\">https://juliagizmos.github.io/WebIO.jl/latest/troubleshooting/not-detected/</a></p>' +\n",
       "                '</div>'\n",
       "            );\n",
       "        return;\n",
       "    }\n",
       "    WebIO.mount(\n",
       "        document.querySelector('[data-webio-mountpoint=\"7093032502373938503\"]'),\n",
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       "    Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Scope(Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :label), Any[\"y\"], Dict{Symbol, Any}(:className => \"interact \", :style => Dict{Any, Any}(:padding => \"5px 10px 0px 10px\")))], Dict{Symbol, Any}(:className => \"interact-flex-row-left\")), Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :input), Any[], Dict{Symbol, Any}(:max => 21, :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}()))], Dict{Symbol, Any}(:className => \"interact-flex-row-center\")), Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :p), Any[], Dict{Symbol, Any}(:attributes => Dict(\"data-bind\" => \"text: formatted_val\")))], Dict{Symbol, Any}(:className => \"interact-flex-row-right\"))], Dict{Symbol, Any}(:className => \"interact-flex-row interact-widget\")), Dict{String, Tuple{Observables.AbstractObservable, Union{Nothing, Bool}}}(\"changes\" => (Observable{Int64} with 1 listeners. Value:\n",
       "0, nothing), \"index\" => (Observable{Any} with 2 listeners. Value:\n",
       "21, nothing)), Set{String}(), nothing, Asset[Asset(\"js\", \"knockout\", \"/home/tisaac/.julia/packages/Knockout/3gjS1/src/../assets/knockout.js\"), Asset(\"js\", \"knockout_punches\", \"/home/tisaac/.julia/packages/Knockout/3gjS1/src/../assets/knockout_punches.js\"), Asset(\"js\", nothing, \"/home/tisaac/.julia/packages/InteractBase/o1I6G/src/../assets/all.js\"), Asset(\"css\", nothing, \"/home/tisaac/.julia/packages/InteractBase/o1I6G/src/../assets/style.css\"), Asset(\"css\", nothing, \"/home/tisaac/.julia/packages/Interact/pVyN2/src/../assets/bulma_confined.min.css\")], Dict{Any, Any}(\"changes\" => Any[WebIO.JSString(\"(function (val){return (val!=this.model[\\\"changes\\\"]()) ? (this.valueFromJulia[\\\"changes\\\"]=true, this.model[\\\"changes\\\"](val)) : undefined})\")], \"index\" => Any[WebIO.JSString(\"(function (val){return (val!=this.model[\\\"index\\\"]()) ? (this.valueFromJulia[\\\"index\\\"]=true, this.model[\\\"index\\\"](val)) : undefined})\")]), WebIO.ConnectionPool(Channel{Any}(32), Set{AbstractConnection}(), Condition(Base.InvasiveLinkedList{Task}(Task (runnable) @0x00007f89cec37540, Task (runnable) @0x00007f89cec37540), Base.AlwaysLockedST(1))), WebIO.JSString[WebIO.JSString(\"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(\\n                element,\\n                {\\n                    value: stringified,\\n                    valueUpdate: allBindings.get('valueUpdate'),\\n                },\\n                context,\\n            );\\n        }\\n    };\\n    var json_data = {\\\"formatted_vals\\\":[\\\"-1.0\\\",\\\"-0.9\\\",\\\"-0.8\\\",\\\"-0.7\\\",\\\"-0.6\\\",\\\"-0.5\\\",\\\"-0.4\\\",\\\"-0.3\\\",\\\"-0.2\\\",\\\"-0.1\\\",\\\"0.0\\\",\\\"0.1\\\",\\\"0.2\\\",\\\"0.3\\\",\\\"0.4\\\",\\\"0.5\\\",\\\"0.6\\\",\\\"0.7\\\",\\\"0.8\\\",\\\"0.9\\\",\\\"1.0\\\"],\\\"changes\\\":WebIO.getval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"10273822401361527462\\\",\\\"id\\\":\\\"1449421620076096789\\\",\\\"type\\\":\\\"observable\\\"}),\\\"index\\\":WebIO.getval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"10273822401361527462\\\",\\\"id\\\":\\\"6471199102608208031\\\",\\\"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\\\":\\\"10273822401361527462\\\",\\\"id\\\":\\\"1449421620076096789\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"10273822401361527462\\\",\\\"id\\\":\\\"6471199102608208031\\\",\\\"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/46ca1ae6128ce5cef46ae6c83c10cbc27a02b7d5-knockout.js\\\"},{\\\"name\\\":\\\"knockout_punches\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/ae9f554c0387f76781e87553335a640c2663983d-knockout_punches.js\\\"}],\\\"type\\\":\\\"async_block\\\"})).then((imports) => handler.apply(this, imports));\\n}\\n\")])], Dict{Symbol, Any}(:className => \"field interact-widget\")))\n",
       "  Observable{Plots.Plot{Plots.GRBackend}} with 0 listeners. Value:\n",
       "Plot{Plots.GRBackend() n=7})"
      ]
     },
     "execution_count": 6,
     "metadata": {
      "application/vnd.webio.node+json": {
       "kernelId": "a949b378-bb8f-43ca-9cf2-3128dafeda22"
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "if STATIC_LECTURE\n",
    "    draw(plot_projections(1.0,1.0,-1.0,0.1))\n",
    "else\n",
    "    theta=widget(-2π:0.05π:2π, value=-1.0, label=\"θ\")\n",
    "    phi=widget(-2π:0.05π:2π, value=0.1, label=\"ϕ\")\n",
    "    x=widget(-1.0:0.1:1.0, value=1.0, label=\"x\")\n",
    "    y=widget(-1.0:0.1:1.0, value=1.0, label=\"y\")\n",
    "    iplot = map(plot_projections,x,y,theta,phi)\n",
    "    vbox(hbox(theta,phi,x,y),iplot)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "38b5b7b1",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Given vectors $u$ and $v$, what choice of $\\alpha$ makes\n",
    "\n",
    "$$ P = \\alpha u v^* $$\n",
    "\n",
    "a projector?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12705c98",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Orthogonal Projectors\n",
    "\n",
    "A projector $P$ is an _orthogonal projector_ if\n",
    "\n",
    "$$ \\mathrm{range}(P) \\perp \\mathrm{range}(I-P). $$\n",
    "\n",
    "($S \\perp T$: every vector in $S$ or orthogonal to every vector in $T$.)\n",
    "\n",
    "We mostly use orthogonal projectors: we call non-orthogonal projectors _oblique projectors_."
   ]
  },
  {
   "attachments": {
    "ortho-projector.svg": {
     "image/svg+xml": [
      "<?xml version='1.0' encoding='UTF-8'?>
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   "source": [
    "![ortho-projector.svg](attachment:ortho-projector.svg)"
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    "### Algebraic definition of orthogonal projector\n",
    "\n",
    "A projector $P$ is orthogonal if and only if\n",
    "\n",
    "$$P = P^*.$$"
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    "### Orthogonal projector $\\neq$ orthogonal AKA unitary"
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   "source": [
    "**Orthogonal Projector**\n",
    "\n",
    "$$ P = P^2 $$\n",
    "\n",
    "$$ P = P^* $$"
   ]
  },
  {
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   "source": [
    "**Orthogonal / Unitary**\n",
    "\n",
    "$$ P P^* = I $$\n",
    "\n",
    "$$ P^* P = I $$"
   ]
  },
  {
   "cell_type": "markdown",
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   "metadata": {},
   "source": [
    "Only one matrix is both..."
   ]
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   "source": [
    "$$I$$\n",
    "\n",
    "The only full-rank projector; the only idempotent orthogonal matrix."
   ]
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    "### The definitions are equivalent\n",
    "\n",
    "**T&B Theorem 6.1.** If $P$ is a projector, then $P = P^*$ if and only if $\\mathrm{range}(P) \\perp \\mathrm{range}(I-P)$.\n",
    "\n",
    "_Proof._"
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    "$\\Rightarrow$: If $P = P^*$, then\n",
    "\n",
    "$$(Px)^*((I - P)y) = x^*(P^*(I-P) y) = x^* (P(I-P) y) = x^* 0 = 0.$$"
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   "source": [
    "$\\Leftarrow$: If $\\mathrm{range}(P) \\perp \\mathrm{range}(I-P)$, then\n",
    "\n",
    "$$\n",
    "(Px)^* y = (Px)^*(Py + (I-P)y) = (Px)^*Py + \\underbrace{(Px)^*(I-P)y}_{0} = x^*(P^*P)y\n",
    "$$\n",
    "\n",
    "Therefore $$P^* = \\underbrace{P^* P}_{\\text{(hermitian)}}.$$"
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    "### Exercise\n",
    "\n",
    "**T&B Exercise 6.1.** If $P$ is an orthogonal projector, then $I - 2P$ is unitary."
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    "## Projection with an orthonormal basis\n",
    "\n",
    "**Theorem.** Let $P \\in \\mathbb{C}^{m\\times m}$ be an orthogonal projector with rank $n \\leq m$.  Let the columns of $\\hat Q\\in\\mathbb{C}^{m \\times n}$ form an orthonormal basis of $\\mathrm{range}(P)$.  Then\n",
    "\n",
    "$$P = \\hat Q \\hat Q^*.$$\n",
    "\n",
    "_Proof._ $P$ is defined by $Pv = v$ for all $v\\in \\mathrm{range}(P)$ and $Pv = 0$ for all $v \\perp \\mathrm{range}(P)$.\n",
    "\n",
    "$\\hat Q \\hat Q^*$ has these properties, so they are equal."
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   "id": "8cd47a8d",
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   "source": [
    "The choice of orthonormal basis of $\\mathrm{range}(P)$ doesn't matter.\n",
    "\n",
    "If the columns of $\\hat Q$ are orthonormal, and the columns of $\\hat W$ are orthonormal and span the same subspace,\n",
    "\n",
    "$$\\hat Q \\hat Q^* = \\hat W \\hat W^*.$$"
   ]
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 5; n = 3;\n",
    "# n random orthonormal columns\n",
    "Q̂ = qr(rand(m, n)).Q[:,1:n];\n",
    "P = Q̂ * Q̂';\n",
    "draw(imshow(P))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "4f1787cb",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
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      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute another basis for range(P) via SVD\n",
    "U, σ, V = svd(P)\n",
    "Ŵ = U[:,1:3]\n",
    "# Q̂ ≠ Ŵ ...\n",
    "draw(plot(imshow(Q̂),imshow(Ŵ),layout=(1,2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "2975bedd",
   "metadata": {
    "cell_style": "center"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4701388995066137e-15"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "norm(P - Ŵ * Ŵ') # ... but Q*Q' = P = W * W'"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96eb1c6b",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Like an outer product, if $n \\ll m$, it is better to store $\\hat{Q}$ than to store $P$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "692470fc",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "(Or if $(m -n) \\ll n$, store the complement $\\hat{Q}^c$ that spans $\\mathrm{range}(I-P)$.)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ee92801",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "### Special case: projection onto span of one vector\n",
    "\n",
    "If $u$ is a unit vector, the projector onto $\\mathrm{span}\\{u\\}$ is $$P_u = u u^*.$$\n",
    "\n",
    "The complementary projector is $$P_{\\perp u} = I - u u^*.$$\n",
    "\n",
    "($P_{\\perp u} v$ \"removes the $u$ component of $v$\".)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa3f8e2c",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "For an arbitrary vector $a$, we don't have to normalize before defining the orthogonal projectors:\n",
    "\n",
    "$$P_a = \\frac{a a^*}{a^* a}, \\quad P_{\\perp a} = I - \\frac{a a^*}{a^* a}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ff16d81",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "subslide"
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   },
   "source": [
    "### Projection with an arbitirary basis\n",
    "\n",
    "The case is the same for larger subspaces: given $\\hat{A}$ with linearly independent columns that span $\\mathrm{range}(P)$,\n",
    "\n",
    "$$P_{\\hat{A}} = \\hat{A} (\\hat{A}^* \\hat{A})^{-1} \\hat{A}^*$$\n",
    "\n",
    "$$P_{\\perp \\hat{A}} = I - \\hat{A} (\\hat{A}^* \\hat{A})^{-1} \\hat{A}^*.$$\n",
    "\n",
    "(Notice symmetry of $P$ and all values in $P \\leq 1$.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "6e49be7f",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Projector error 4.381630010759598e-16\n",
      "Range error 1.6346297495750493e-15\n",
      "Hermitian error 4.0639252874588374e-16\n"
     ]
    },
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      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 10; n = 5;\n",
    "\n",
    "Â = randn(m, n)\n",
    "\n",
    "# P = (Â / (Â' * Â)) * Â' # apply inverse on right\n",
    "P = Â *((Â' * Â) \\ Â') # apply inverse on left\n",
    "println(\"Projector error $(norm(P - P*P, 2))\")\n",
    "println(\"Range error $(norm((I-P)*Â, 2))\")\n",
    "println(\"Hermitian error $(norm(P - P', 2))\")\n",
    "draw(imshow(P))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c4e3578c",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### What about oblique projectors?\n",
    "\n",
    "Let projector $P$ have rank $n$, let $\\hat{A} \\in \\mathbb{C}^{m \\times n}$ have columns that span $\\mathrm{range}(A)$, \n",
    "and let $\\hat{B} \\in \\mathbb{C}^{m \\times (m-n)}$ span $\\mathrm{null}(P)$.\n",
    "\n",
    "Define $C = [\\hat{A} | \\hat{B}]$, which is necessarily nonsingular.\n",
    "\n",
    "Let $\\hat{Z}$ be the first $n$ rows of $C^{-1}$.\n",
    "\n",
    "$$P = \\hat{A} \\hat{Z}.$$\n",
    "\n",
    "(Notice $P$ is not symmetric and can have values $> 1$.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "8ce1b34e",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Projector error 1.1042047974356696e-11\n",
      "Range error 1.266530629480315e-13\n",
      "Nullspace error 1.3792425453875028e-13\n"
     ]
    },
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 10; n = 5;\n",
    "\n",
    "Â = randn(m, n)\n",
    "B̂ = randn(m, m - n)\n",
    "\n",
    "C = [Â B̂]\n",
    "\n",
    "Z = C^-1\n",
    "Ẑ = Z[1:n,:]\n",
    "\n",
    "P = Â * Ẑ;\n",
    "println(\"Projector error $(norm(P - P*P, 2))\")\n",
    "println(\"Range error $(norm((I-P)*Â, 2))\")\n",
    "println(\"Nullspace error $(norm(P*B̂, 2))\")\n",
    "draw(imshow(P))"
   ]
  },
  {
   "attachments": {
    "projector-hw.svg": {
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   "source": [
    "### Notes for HW2, problem 1\n",
    "\n",
    "Proving ($P$ is nonzero orthogonal projector) $\\Rightarrow$ ($\\|P\\|_2 = 1$): straightforward application of definitions.\n",
    "\n",
    "Proving ($\\|P\\|_2 = 1$) $\\Rightarrow$ ($P$ is a nonzero orthogonal projector): Suppose $P$ is oblique.  Let $\\tilde{P}$ be an orthogonal projector with the same nullspace, and suppose $v \\in \\mathrm{range}(\\tilde{P})$.  In the diagram it is clear that\n",
    "$\\|Pv\\|_2 > \\|v\\|_2.$  Use the definitions and the Pythagorean theorem to show this is always the case.\n",
    "\n",
    "![projector-hw.svg](attachment:projector-hw.svg)"
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   "id": "d10c18f3",
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     "slide_type": "slide"
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   },
   "source": [
    "## QR Factorization\n",
    "\n",
    "> One algorithmic idea in numerical linear algebra is more important than all the others: QR factorization."
   ]
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  {
   "cell_type": "markdown",
   "id": "3759e6a0",
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    "### Want bases for successive (nested) spaces spanned by the columns of $A$\n",
    "\n",
    "$$\n",
    "\\mathrm{span}\\{a_1\\} \\subseteq\n",
    "\\mathrm{span}\\{a_1, a_2\\} \\subseteq\n",
    "\\mathrm{span}\\{a_1, a_2, a_3\\} \\subseteq \\dots\n",
    "$$"
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  {
   "cell_type": "markdown",
   "id": "9c7222a2",
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   "source": [
    "### Use nested orthogonal bases\n",
    "\n",
    "Given $A \\in \\mathbb{C}^{m \\times n}$ ($m \\geq n$) that is _full rank_, want to find orthonormal vectors $\\{q_1, \\dots, q_n\\}$\n",
    "such that\n",
    "\n",
    "$$\n",
    "\\mathrm{span}\\{q_1, \\dots, q_j\\} =\n",
    "\\mathrm{span}\\{a_1, \\dots, a_j\\}, \\quad j = 1, \\dots, n\n",
    "$$"
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  {
   "cell_type": "markdown",
   "id": "fe326dca",
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   "source": [
    "### This is a triangular relationship\n",
    "\n",
    "$$\n",
    "\\left[\n",
    "\\begin{array}{c|c|c|c}\n",
    "& & & \\\\\n",
    "& & & \\\\\n",
    "a_1 & a_2 & \\dots & a_n \\\\\n",
    "& & & \\\\\n",
    "& & &\n",
    "\\end{array}\n",
    "\\right]\n",
    "=\n",
    "\\left[\n",
    "\\begin{array}{c|c|c|c}\n",
    "& & & \\\\\n",
    "& & & \\\\\n",
    "q_1 & q_2 & \\dots & q_n \\\\\n",
    "& & & \\\\\n",
    "& & &\n",
    "\\end{array}\n",
    "\\right]\n",
    "\\begin{bmatrix}\n",
    "r_{11} & r_{12} & \\dots & r_{1n} \\\\\n",
    "       & r_{22} & & r_{2n} \\\\\n",
    "       &        & \\ddots & \\vdots \\\\\n",
    "                & & & r_{nn}\n",
    "\\end{bmatrix}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0dae7d08",
   "metadata": {
    "slideshow": {
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   "source": [
    "### As a matrix equation: the Reduced QR Factorization of $A$\n",
    "\n",
    "$$ A = \\hat{Q} \\hat R,$$\n",
    "\n",
    "$\\hat{Q} \\in \\mathbb{C}^{m \\times n}$ with orthonormal columns, $\\hat R \\in \\mathbb{C}^{n \\times n}$ upper triangular."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee85d23c",
   "metadata": {
    "slideshow": {
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   },
   "source": [
    "### Full QR Factorization: Expand $\\hat{Q}$ to a unitary matrix"
   ]
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    "The extra columns added to $\\hat{Q}$ are\n",
    "orthogonal to $\\mathrm{range}(A)$.\n",
    "\n",
    "Thisis analogous to full vs. reduced SVD.\n",
    "\n",
    "$ $\n",
    "\n",
    "$ $\n",
    "$ $"
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   },
   "cell_type": "markdown",
   "id": "781eb867",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "![full-qr.svg](attachment:full-qr.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5a10f117",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The curious case of julia's `qr`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "8c16583f",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(10, 10)"
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     },
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    },
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      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 10; n = 5;\n",
    "\n",
    "A = randn(m, n)\n",
    "Q, R = qr(A)\n",
    "display(size(Q))\n",
    "draw(imshow(Q))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "8a285be1",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5, 5)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display(size(R))\n",
    "draw(imshow(R))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9caf557",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### It appears to be a half-reduced / half-full QR factorization"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2dc17083",
   "metadata": {},
   "source": [
    "... but we can multiply $Q$ and $\\hat{R}$, even though the shapes don't match?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "81ef4dec",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2665821846872752e-15"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(A - Q * R, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83c91bb7",
   "metadata": {},
   "source": [
    "In fact we can multiply $Q$ by any matrix with $m$ rows _and_ any matrix with $n$ rows?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "b8898430",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.843992188335864"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "4.56133840610496"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(norm(Q * randn(10,7), 2))\n",
    "display(norm(Q * randn(5,7), 2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5bb07eb",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### julia `qr`'s `Q` is an object that can pretend to be $Q$ or $\\hat{Q}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "ace36081",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}}"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(Q)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b8ad144",
   "metadata": {},
   "source": [
    "- Extract the first $n$ columns if you need a plain matrix of $\\hat{Q}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "1a049648",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix{Float64} (alias for Array{Float64, 2})"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(Q[:,1:n])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4143384",
   "metadata": {},
   "source": [
    "- Extract all the columns if you need a plain matrix of $Q$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "c03d69b6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix{Float64} (alias for Array{Float64, 2})"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(Q[:,:])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd7eadb8",
   "metadata": {},
   "source": [
    "- If you only need to multiply by $Q$ or $Q^*$, just use `Q`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c4dd1df",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Notes on HW2, problem 2\n",
    "\n",
    "- Each plane in 2D has a _normal vector_, orthogonal to every vector in the plane.\n",
    "- So a line in the intersection of two planes is orthogonal to _both_ planes' normal vectors.\n",
    "- This is a good, short program to get started with writing julia."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "28e956e3",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## QR by Projections\n",
    "\n",
    "### Reminder of the original purpose: nested orthogonal bases of the nested column spaces of $A$ \n",
    "\n",
    "$$A = \\hat{Q}\\hat{R}, \\quad A \\in \\mathbb{C}^{m \\times n}\\ (m \\geq n), \\quad \\hat Q \\in \\mathbb{C}^{m \\times n}\\ (\\hat Q^* \\hat Q = I), \\quad \\hat R \\in \\mathbb{C}^{n \\times n}\\ (r_{ij} = 0, i > j).$$\n",
    "\n",
    "$$\\mathrm{span}\\{q_1, \\dots, q_j\\} = \\mathrm{span}\\{a_1, \\dots, a_j\\}$$\n",
    "\n",
    "For now suppose $A$ has rank $n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "f889dd5a",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 10; n = 5;\n",
    "\n",
    "A = randn(m, n);\n",
    "\n",
    "draw(imshow(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19acff03",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Base case: $j=1$\n",
    "\n",
    "$$\\mathrm{span}\\{a_1\\} = \\mathrm{span}\\{q_1\\}$$\n",
    "\n",
    "The first column of $A = \\hat{Q} \\hat{R}$ is\n",
    "\n",
    "$$\n",
    "a_1 = q_1 r_{11}\n",
    "$$\n",
    "\n",
    "Just set\n",
    "\n",
    "$$q_1 = \\frac{a_1}{\\|a_1\\|}, \\quad r_{11} = \\|a_1\\| = q_1^* a_1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "c16810e5",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
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     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q̂ = zeros(m, n); R̂ = zeros(n, n)\n",
    "\n",
    "R̂[1,1] = norm(A[:,1])\n",
    "Q̂[:,1] = A[:,1] / R̂[1,1]\n",
    "draw(plot(imshow(A), imshow(Q̂), imshow(R̂), layout=(1,3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "74885795",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Consider $j=2$\n",
    "\n",
    "Have\n",
    "\n",
    "$$\\mathrm{span}\\{q_1\\} = \\mathrm{span}\\{a_1\\},$$\n",
    "\n",
    "want\n",
    "\n",
    "$$\\mathrm{span}\\{q_1, q_2\\} = \\mathrm{span}\\{a_1, a_2\\}, \\quad a_2 = r_{12} q_1 + r_{22} q_{22}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a23d7484",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Note that $$\\mathrm{span}\\{a_1, a_2\\} = \\mathrm{span}\\{q_1, a_2\\},$$ so we want\n",
    "\n",
    "$$q_2 \\in \\mathrm{span}\\{q_1, a_2\\}, \\quad q_2 \\perp \\mathrm{span}\\{q_1\\}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de841f75",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Project the $q_1$ component out of $a_2$ to create $q_2$\n",
    "\n",
    "$$P_1 = q_1 q_1^*, \\quad \\tilde q_2 = (I - P_1) a_2.$$\n",
    "\n",
    "But $\\tilde q_2$ isn't unit length, so normalize it:\n",
    "\n",
    "$$\n",
    "q_2 = \\frac{\\tilde q_2}{\\|\\tilde q_2\\|}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "0ffd7dbb",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1 = Q̂[:,1] * Q̂[:,1]'\n",
    "Q̂[:,2] = (I - P1) * A[:,2]\n",
    "Q̂[:,2] /= norm(Q̂[:,2])\n",
    "\n",
    "draw(plot(imshow(A), imshow(Q̂), imshow(R̂), layout=(1,3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f1e8b180",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Compute the second column of $\\hat{R}$\n",
    "\n",
    "Now we have an orthogonal projector onto $\\mathrm{span}\\{q_1, q_2\\} = \\mathrm{span}\\{a_1, a_2\\},$\n",
    "\n",
    "$$P_{1:2} = \\hat{Q}_{1:2} \\hat{Q}_{1:2}^*.$$\n",
    "\n",
    "Because $a_2 \\in \\mathrm{span}\\{a_1, a_2\\}$, \n",
    "\n",
    "$$a_2 = P_{1:2} a_2 = \\hat{Q}_{1:2}\\underbrace{\\hat{Q}_{1:2}^* a_2}_{r_{1:2,2}}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "f40eb0a0",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R̂[1:2,2] = Q̂[:,1:2]' * A[:,2]\n",
    "draw(plot(imshow(A), imshow(Q̂), imshow(R̂), layout=(1,3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eea0e495",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### General iteration $j$\n",
    "\n",
    "Project the $q_1, \\dots, q_{j-1}$ component out of $a_j$ and normalize:\n",
    "\n",
    "$$\n",
    "\\tilde{q}_j = (I - P_{1:j-1}) a_j, \\quad q_j = \\frac{\\tilde{q}_j}{\\|\\tilde{q}_j\\|}\n",
    "$$\n",
    "\n",
    "The $j$th column of $\\hat{R}$ is\n",
    "\n",
    "$$\n",
    "r_{1:j,j} = \\hat Q_{1:j}^* a_j.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3e5b1bc3",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### In action\n",
    "\n",
    "(Left matrix: $A - \\hat{Q} * \\hat{R}$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "cdb8b7a2",
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Saved animation to \n",
      "│   fn = /home/tisaac/srv/rep/22-course-cse6643/mybinder/tmp.gif\n",
      "└ @ Plots /home/tisaac/.julia/packages/Plots/Xv2qA/src/animation.jl:114\n"
     ]
    },
    {
     "data": {
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\" />"
      ],
      "text/plain": [
       "Plots.AnimatedGif(\"/home/tisaac/srv/rep/22-course-cse6643/mybinder/tmp.gif\")"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q̂[:,:] .= 0.; R̂[:,:] .= 0.;\n",
    "a = Animation()\n",
    "for s in 1:2*n\n",
    "    j = div(s + 1, 2)\n",
    "    title = \"\"\n",
    "    if mod(s, 2) == 1\n",
    "        P = Q̂[:,1:j-1] * Q̂[:,1:j-1]'\n",
    "        Q̂[:,j] = (I - P) * A[:,j]\n",
    "        Q̂[:,j] /= norm(Q̂[:,j])\n",
    "        title = \"Orthogonalize column $j\"\n",
    "    else\n",
    "        R̂[1:j,j] = Q̂[:,1:j]' * A[:,j]\n",
    "        title = \"Q̂ coefficients of column $j\"\n",
    "    end\n",
    "    p = plot(imshow(A - Q̂ * R̂), imshow(Q̂), imshow(R̂), layout=(1,3), plot_title=title)\n",
    "    frame(a, p)\n",
    "end\n",
    "gif(a, fps=0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c0a96ac",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### For demonstration purposes only\n",
    "\n",
    "Next time we will see practical implementations of this procedure"
   ]
  }
 ],
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