lec05static.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "25289497",
   "metadata": {
    "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 = true;"
   ]
  },
  {
   "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": [
    {
     "data": {
      "application/vnd.webio.node+json": {
       "children": [],
       "instanceArgs": {
        "namespace": "html",
        "tag": "div"
       },
       "nodeType": "DOM",
       "props": {},
       "type": "node"
      },
      "text/html": [
       "<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()"
      ]
     },
     "metadata": {
      "application/vnd.webio.node+json": {
       "kernelId": null
      }
     },
     "output_type": "display_data"
    },
    {
     "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": [
    "### 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",
    "$$ P^2 v = P v  \\quad \\Leftrightarrow \\quad P (\\lambda v) = \\lambda v\n",
    "\\quad \\Leftrightarrow \\quad P v = v $$"
   ]
  },
  {
   "attachments": {
    "projector.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "def61a7a",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Intuition: \"projected\" like a shadow on a wall\n",
    "\n",
    "![projector.png](attachment:projector.png)"
   ]
  },
  {
   "attachments": {
    "projector2.png": {
     "image/png": 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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.png](attachment:projector2.png)"
   ]
  },
  {
   "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"
    }
   },
   "source": [
    "### Complementary Projectors\n",
    "\n",
    "If $P$ is a projector, $I - P$ is also a projector, the _complementary projector_:\n",
    "\n",
    "$$(I - P)^2 = I - 2 P + P^2 = I - 2 P + P = I - P.$$"
   ]
  },
  {
   "attachments": {
    "projector3.png": {
     "image/png": 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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.png](attachment:projector3.png)"
   ]
  },
  {
   "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;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "d6744388",
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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.png": {
     "image/png": 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    }
   },
   "cell_type": "markdown",
   "id": "a9a936d8",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "![ortho-projector.png](attachment:ortho-projector.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a850540c",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Algebraic definition of orthogonal projector\n",
    "\n",
    "A projector $P$ is orthogonal if and only if\n",
    "\n",
    "$$P = P^*.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52d34eee",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Orthogonal projector $\\neq$ orthogonal AKA unitary"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4ad51c0",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "**Orthogonal Projector**\n",
    "\n",
    "$$ P = P^2 $$\n",
    "\n",
    "$$ P = P^* $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1954cba4",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "**Orthogonal / Unitary**\n",
    "\n",
    "$$ P P^* = I $$\n",
    "\n",
    "$$ P^* P = I $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43706d11",
   "metadata": {},
   "source": [
    "Only one matrix is both..."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23614d07",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$I$$\n",
    "\n",
    "The only full-rank projector; the only idempotent orthogonal matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c1c6e56",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 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._"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02caa785",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$\\Rightarrow$: If $P = P^*$, then\n",
    "\n",
    "$$(Px)^*((I - P)y) = x^*(P^*(I-P) y) = x^* (P(I-P) y) = x^* 0 = 0.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb855800",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "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 $$P^* = \\underbrace{P^* P}_{\\text{(hermitian)}}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f97d45c",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "**T&B Exercise 6.1.** If $P$ is an orthogonal projector, then $I - 2P$ is unitary."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "937d3fcc",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 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."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8cd47a8d",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "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",
    "$$\\hat Q \\hat Q^* = \\hat W \\hat W^*.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "c2db804e",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": [
    {
     "data": {
      "image/png": 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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": [
       "7.927363627673169e-16"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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 $m \\ll n$, 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"
    }
   },
   "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": [
    {
     "data": {
      "image/png": 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"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Projector error 4.982651588016061e-16\n",
      "Range error 1.2420993031112908e-15\n",
      "Hermitian error 4.880711628404164e-16\n"
     ]
    }
   ],
   "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) \\ A') # 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": 11,
   "id": "8ce1b34e",
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "image/png": "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"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Projector error 3.0177612764965533e-15\n",
      "Range error 3.120015185807418e-15\n",
      "Nullspace error 3.0595897420912996e-15\n"
     ]
    }
   ],
   "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 $(norm(P*B̂, 2))\")\n",
    "draw(imshow(P))"
   ]
  },
  {
   "attachments": {
    "projector-hw.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "34d33f39",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Notes for HW2, problem 1\n",
    "\n",
    "Proving ($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.png](attachment:projector-hw.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d10c18f3",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## QR Factorization\n",
    "\n",
    "> One algorithmic idea in numerical linear algebra is more important than all the others: QR factorization."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3759e6a0",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Want bases for successive (nested) spaces spanned by the columns of $A$\n",
    "\n",
    "$$\n",
    "\\mathrm{span}\\{a_1\\} \\subseteq\n",
    "\\mathrm{span}\\{a_1, a_2\\} \\subseteq\n",
    "\\mathrm{span}\\{a_1, a_2, a_3\\} \\subseteq \\dots\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c7222a2",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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",
    "$$\n",
    "\\mathrm{span}\\{q_1, \\dots, q_j\\} =\n",
    "\\mathrm{span}\\{a_1, \\dots, a_j\\}, \\quad j = 1, \\dots, n\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe326dca",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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": {
     "slide_type": "subslide"
    }
   },
   "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": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Full QR Factorization: Expand $\\hat{Q}$ to a unitary matrix"
   ]
  },
  {
   "attachments": {
    "reduced-qr.png": {
     "image/png": 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    }
   },
   "cell_type": "markdown",
   "id": "b9060ce2",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "![reduced-qr.png](attachment:reduced-qr.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5a679858",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "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",
    "$ $\n",
    "\n",
    "$ $\n",
    "$ $"
   ]
  },
  {
   "attachments": {
    "full-qr.png": {
     "image/png": 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    }
   },
   "cell_type": "markdown",
   "id": "781eb867",
   "metadata": {
    "cell_style": "split"
   },
   "source": [
    "![full-qr.png](attachment:full-qr.png)"
   ]
  },
  {
   "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)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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"
    },
    {
     "data": {
      "image/png": "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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": [
       "1.5408419453419898e-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": [
       "8.35630883728406"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "5.7326887746110335"
      ]
     },
     "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": [
    {
     "data": {
      "image/png": "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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": [
    {
     "data": {
      "image/png": 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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": [
    {
     "data": {
      "image/png": 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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": [
    {
     "data": {
      "image/png": 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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": {
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   "source": [
    "### For demonstration purposes only\n",
    "\n",
    "Next time we will see practical implementations of this procedure"
   ]
  }
 ],
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