{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Intro to Generalized Stability Theory\n",
    "\n",
    "\n",
    "#### Navid C. Constantinou\n",
    "#### RSES, ANU, 2018"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Linear Systems\n",
    "\n",
    "Consider the linear system that describes perturbations about a basic state,\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d} \\boldsymbol{\\phi}}{\\mathrm{d}t}  = \\mathbb{A}\\,\\boldsymbol{\\phi},\\quad \\boldsymbol{\\phi}(t_0)=\\boldsymbol{\\phi}_0. \\tag{1}\n",
    "\\end{align}\n",
    "\n",
    "Above, $\\boldsymbol{\\phi}$ is the state vector that collectively describes the perturbations about the basic state and  $\\mathbb{A}$ is a linear operator (which implicitly depends on the basic state).\n",
    "\n",
    "We are interested on how the energy of the perturbations evolves with time.\n",
    "\n",
    "Energy is a quadratic quantity and is, usually, defined through an inner product. Let's begin for simplicity with the case in which the perturbation energy is given through the inner product:\n",
    "\n",
    "\\begin{align}\n",
    "E(t) = \\big(\\boldsymbol{\\phi}(t), \\boldsymbol{\\phi}(t)\\big). \\tag{2}\n",
    "\\end{align}\n",
    "\n",
    "For example, for a finite-dimensional state space, e.g., $\\mathbb{C}^{n\\times 1}$, the inner product can be simply the Euclidean inner product \n",
    "\n",
    "\\begin{align}\n",
    "(\\boldsymbol{\\psi}, \\boldsymbol{\\phi}) \\equiv \\boldsymbol{\\psi}^\\dagger \\boldsymbol{\\phi} = \\sum_{j=1}^n \\psi_j^* \\phi_j. \\tag{3}\n",
    "\\end{align}\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Motivation\n",
    "\n",
    "Some questions that we will try to answer are:\n",
    "\n",
    "\n",
    "1. Can perturbations grow if the linear operator has no modal instabilities?\n",
    "\n",
    "2. What is the best way to initiate linear system (1) so that we get maximum energy for $t\\to\\infty$? Specifically, what's the initial condition of unit norm $\\boldsymbol{\\phi}_0$, with $\\|\\boldsymbol{\\phi}_0\\|=1$ and how much can energy grow?\n",
    "\n",
    "3. If we are interested in maximum energy growth for very short times $t\\ll 1$ do we have to initiate the linear system differently?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Linear Systems\n",
    "\n",
    "The linear system (1) has solution:\n",
    "\n",
    "\\begin{align}\n",
    "\\boldsymbol{\\phi}(t) = \\mathbb{P}(t,t_0)\\,\\boldsymbol{\\phi}_0 , \\tag{4}\n",
    "\\end{align}\n",
    "\n",
    "where $\\mathbb{P}(t,t_0)$ is a *linear* operator that maps the state at time $t_0$ to the state at time $t$. Linear map $\\mathbb{P}$ is called the *propagator*. \n",
    "\n",
    "When the linear operator in (1) is time-independent then the propagator is nothing else than\n",
    "\n",
    "\\begin{align}\n",
    "\\mathbb{P}(t,t_0) = \\exp{(\\mathbb{A}(t-t_0))} , \\tag{5}\n",
    "\\end{align}\n",
    "\n",
    "defined as\n",
    "\n",
    "\\begin{align}\n",
    "\\exp{\\big(\\mathbb{A}(t-t_0)\\big)} = \\sum_{n=0}^{\\infty} \\frac{\\mathbb{A}^n(t-t_0)^n}{n!} . \\tag{6}\n",
    "\\end{align}\n",
    "\n",
    "\n",
    "**Exercise**: Verify that expression (4) with $\\mathbb{P}$ given by (5) solves (1).\n",
    "\n",
    "From hereafter let's take $t_0=0$ for simplicity.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Adjoint and self-adjoint operators\n",
    "\n",
    "**Definition 1 (Adjoint operator)**: The adjoint operator of $\\mathbb{A}$ is denoted as $\\mathbb{A}^\\dagger$ and is defined through:\n",
    "\n",
    "\\begin{align}\n",
    "(\\boldsymbol{\\psi}, \\mathbb{A}\\boldsymbol{\\phi}) = (\\mathbb{A}^\\dagger\\boldsymbol{\\psi}, \\boldsymbol{\\phi})\\ \\ \\mathit{for}\\ \\mathit{every}\\ \\boldsymbol{\\phi},\\boldsymbol{\\psi} \\text{ in the vector space}. \\tag{8}\n",
    "\\end{align}\n",
    "\n",
    "Note that the adjoint operator depends on the inner product!\n",
    "\n",
    "**Definition 2 (Self-adjoint operator)**: An operator $\\mathbb{A}$ is called *self-adjoint* or *Hermitian* if and only if $\\mathbb{A}=\\mathbb{A}^\\dagger$.\n",
    "\n",
    "**Theorem I**: Self-adjoint operators have real eigenvalues.\n",
    "\n",
    "*Proof*: If $\\lambda$ is an eigenvalue of $\\mathbb{A}$ and $\\boldsymbol{u}$ the corresponding eigenvector, $\\mathbb{A}\\,\\boldsymbol{u} = \\lambda\\,\\boldsymbol{u}$. By forming:\n",
    "\n",
    "\\begin{align*}\n",
    "(\\boldsymbol{u}, \\mathbb{A}\\boldsymbol{u}) = (\\boldsymbol{u}, \\lambda \\boldsymbol{u}) = \\lambda (\\boldsymbol{u}, \\boldsymbol{u}).\n",
    "\\end{align*}\n",
    "\n",
    "On the other hand, using that $\\mathbb{A} = \\mathbb{A}^\\dagger$ we have\n",
    "\n",
    "\\begin{align*}\n",
    "(\\boldsymbol{u}, \\mathbb{A}\\boldsymbol{u}) = (\\mathbb{A}^\\dagger \\boldsymbol{u}, \\boldsymbol{u}) = (\\mathbb{A} \\boldsymbol{u}, \\boldsymbol{u}) = (\\lambda \\boldsymbol{u}, \\boldsymbol{u}) =\\lambda^* (\\boldsymbol{u}, \\boldsymbol{u}).\n",
    "\\end{align*}\n",
    "\n",
    "From these we get that $(\\lambda-\\lambda^*)(\\boldsymbol{u}, \\boldsymbol{u})=0$ and since $\\boldsymbol{u}\\ne 0$ (why is that?) we get that $\\lambda=\\lambda^*$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Normal operators\n",
    "\n",
    "**Definition 3 (Normal operator)**: An operator $\\mathbb{A}$ is called *normal* if and only if it commutes with its adjoint: $\\mathbb{A}\\mathbb{A}^\\dagger=\\mathbb{A}^\\dagger\\mathbb{A}$.\n",
    "\n",
    "**Corrolary**: Self-adjoint operators are necessarily normal.\n",
    "\n",
    "**Theorem II**: Normal operators have orthonormal eigenvectors.\n",
    "\n",
    "The proof for this one is a bit cumbersome. We will prove it here for the case of self-adjoint operators, which are a subset of normal operators.\n",
    "\n",
    "*Proof*: Consider a non-degenerate operators. Assume $\\boldsymbol{u}_1$ and $\\boldsymbol{u}_2$ are two eigenvectors of $\\mathbb{A}$ that correspond to eigenvalues $\\lambda_1,\\lambda_2$ ($\\lambda_1\\ne \\lambda_2$). Then we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\big(\\boldsymbol{u}_1,\\mathbb{A}\\boldsymbol{u}_2\\big) = \\lambda_2 \\big(\\boldsymbol{u}_1,\\boldsymbol{u}_2\\big).\n",
    "\\end{align*}\n",
    "\n",
    "But also we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\big(\\boldsymbol{u}_1,\\mathbb{A}\\boldsymbol{u}_2\\big) = \\big(\\mathbb{A}^\\dagger\\boldsymbol{u}_1,\\boldsymbol{u}_2\\big)= \\big(\\mathbb{A}\\boldsymbol{u}_1,\\boldsymbol{u}_2\\big) = \\lambda_1 \\big(\\boldsymbol{u}_1,\\boldsymbol{u}_2\\big).\n",
    "\\end{align*}\n",
    "\n",
    "Combining the two we have $(\\lambda_1-\\lambda_2) \\big(\\boldsymbol{u}_1,\\boldsymbol{u}_2\\big) = 0$ from which it follows that $\\big(\\boldsymbol{u}_1,\\boldsymbol{u}_2\\big) = 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The following code snippet demonstrates the properties of self-adjoint operators."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "using LinearAlgebra, Printf, PyPlot\n",
    "Base.show(io::IO, f::Float64) = @printf(io, \"%1.2f\", f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A=\n",
      "2×2 Array{Float64,2}:\n",
      " -1.50  -0.50\n",
      " -0.50  -1.50 \n",
      " \n",
      "norm(A-A†)= 0.00, eigvalues of A: [-2.00, -1.00], and (u₁, u₂) = 0.00\n",
      " \n",
      "A=\n",
      "2×2 Array{Int64,2}:\n",
      " 0  1\n",
      " 1  0 \n",
      " \n",
      "norm(A-A†)= 0.00, eigvalues of A: [-1.00, 1.00], and (u₁, u₂) = 0.00\n",
      " \n",
      "A=\n",
      "2×2 Array{Complex{Float64},2}:\n",
      " 0.97+0.00im  0.52-0.32im\n",
      " 0.52+0.32im  0.33+0.00im \n",
      " \n",
      "norm(A-A†)= 0.00, eigvalues of A: [-0.04, 1.35], and (u₁, u₂) = 0.00 + 0.00im\n",
      " \n"
     ]
    }
   ],
   "source": [
    "function analyze_operator(A)\n",
    "    S, U = eigen(A); u1, u2 = U[:, 1], U[:, 2]\n",
    "    println(\"A=\"); show(stdout, \"text/plain\", A);\n",
    "    println(\" \"); println(\" \")\n",
    "    println(\"norm(A-A†)= \", norm(A-A'), \", eigvalues of A: \", S, \", and (u₁, u₂) = \", dot(u1, u2));\n",
    "    println(\" \");\n",
    "end\n",
    "\n",
    "A = [-3/2 -1/2; -1/2 -3/2]; analyze_operator(A)\n",
    "A = [0 1; 1 0]; analyze_operator(A)\n",
    "A = rand(2, 2) + im*rand(2,2); A = (A+A')/2; # this is to convert the random A to a self-adjoint one\n",
    "analyze_operator(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Adjoint operators\n",
    "\n",
    "**Theorem III (Properties of the adjoint)**: The adjoint operator $\\mathbb{A}^\\dagger$ has *(i)* eigenvalues which are complex conjugates of the eigenvalues of $\\mathbb{A}$. Furthermore, *(ii)* the eigenvectors of $\\mathbb{A}$ and $\\mathbb{A}^\\dagger$ that *do not* correspond to a complex conjugate eigenvalue pair are orthogonal to each other.\n",
    "\n",
    "*Proof*:\n",
    "\n",
    "*(i)* The eigenvalues $\\lambda$ of $\\mathbb{A}$ solve the characteristic polynomial $\\mathrm{det}(\\mathbb{A}-\\lambda\\mathbb{I})=0$. Using that $\\mathrm{det}{(\\mathbb{A}^\\dagger)} = \\mathrm{det}{(\\mathbb{A})}^*$ we can show that\n",
    "\n",
    "\\begin{align}\n",
    "0 = \\mathrm{det}(\\mathbb{A}-\\lambda\\mathbb{I})^*=\\mathrm{det}\\big[(\\mathbb{A}-\\lambda\\mathbb{I})^\\dagger\\big]=\\mathrm{det}(\\mathbb{A}^\\dagger-\\lambda^*\\mathbb{I}),\n",
    "\\end{align}\n",
    "\n",
    "which implies that the eigenvalues of $\\mathbb{A}^\\dagger$ are $\\lambda^*$, i.e., the complex conjugates of the eigenvalues of $\\mathbb{A}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "*(ii)* Consider now the eigenvector/eigenvalue relations for $\\mathbb{A}$ and $\\mathbb{A}^\\dagger$:\n",
    "\n",
    "\\begin{align*}\n",
    "\\mathbb{A}\\,\\boldsymbol{u}_j = \\lambda_j\\,\\boldsymbol{u}_j\\quad\\text{and}\\quad \n",
    "\\mathbb{A}^\\dagger\\boldsymbol{\\upsilon}_j = \\lambda_j^*\\,\\boldsymbol{\\upsilon}_j. \\tag{10a,b}\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "Take for simplicity that there is no degeneracy. By taking the inner product of (10a) with the eigenvector $\\boldsymbol{\\upsilon}_i$ we have\n",
    "\n",
    "\\begin{align*}\n",
    "(\\boldsymbol{\\upsilon}_i, \\mathbb{A}\\boldsymbol{u}_j) &= (\\boldsymbol{\\upsilon}_i, \\lambda_j\\boldsymbol{u}_j) = \\lambda_j\\,(\\boldsymbol{\\upsilon}_i,\\boldsymbol{u}_j).\n",
    "\\end{align*}\n",
    "\n",
    "But on the other hand, using the definition of $\\mathbb{A}^\\dagger$, \n",
    "\n",
    "\\begin{align*}\n",
    "(\\boldsymbol{\\upsilon}_i, \\mathbb{A}\\boldsymbol{u}_j) &= (\\mathbb{A}^\\dagger\\boldsymbol{\\upsilon}_i, \\boldsymbol{u}_j) = (\\lambda_i^*\\boldsymbol{\\upsilon}_i, \\boldsymbol{u}_j) = \\lambda_i\\,(\\boldsymbol{\\upsilon}_i,\\boldsymbol{u}_j).\n",
    "\\end{align*}\n",
    "\n",
    "These imply that\n",
    "\n",
    "\\begin{align*}\n",
    "(\\lambda_i-\\lambda_j)\\,(\\boldsymbol{\\upsilon}_i,\\boldsymbol{u}_j) = 0,\n",
    "\\end{align*}\n",
    "\n",
    "and since there is no degeneracy (i.e., $\\lambda_i\\ne\\lambda_j$) it follows that\n",
    "\n",
    "\\begin{align*}\n",
    "(\\boldsymbol{\\upsilon}_i,\\boldsymbol{u}_j) = 0\\text{ for }i\\ne j.\\tag{11}\n",
    "\\end{align*}\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The following code snippet demonstrates the properties of the adjoint operators."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A=\n",
      "2×2 Array{Complex{Float64},2}:\n",
      " 0.67+0.48im  0.11+0.82im\n",
      " 0.56+0.20im  0.39+0.59im \n",
      " \n",
      "eigvalues of A: λ₁=0.09 + 0.00im , λ₂=0.97 + 1.06im\n",
      " \n",
      "A†=\n",
      "2×2 Adjoint{Complex{Float64},Array{Complex{Float64},2}}:\n",
      " 0.67-0.48im  0.56-0.20im\n",
      " 0.11-0.82im  0.39-0.59im \n",
      " \n",
      "eigvalues of A†: μ₁0.09 - 0.00im , μ₂=0.97 - 1.06im\n",
      " \n",
      "inner products between eigenvectors of A and of A†\n",
      "(u₁, v₁) = -0.82 - 0.52im, (u₁, v₂) = 0.00 + 0.00im\n",
      "(u₂, v₁) = -0.00 + 0.00im, (u₂, v₂) = 0.82 + 0.52im\n"
     ]
    }
   ],
   "source": [
    "A = rand(2, 2) + im*rand(2,2); \n",
    "\n",
    "S, U = eigen(A); u1 = U[:, 1]; u2 =U[:, 2]\n",
    "Sadj, V = eigen(copy(A')); v1 = V[:, 1]; v2 =V[:, 2]\n",
    "\n",
    "println(\"A=\"); show(stdout, \"text/plain\", A);\n",
    "println(\" \"); println(\" \")\n",
    "println(\"eigvalues of A: λ₁=\", S[1], \" , λ₂=\", S[2]); println(\" \"); \n",
    "\n",
    "println(\"A†=\"); show(stdout, \"text/plain\", A');\n",
    "println(\" \"); println(\" \")\n",
    "println(\"eigvalues of A†: μ₁\", Sadj[1], \" , μ₂=\", Sadj[2]); println(\" \");\n",
    "\n",
    "println(\"inner products between eigenvectors of A and of A†\");\n",
    "println(\"(u₁, v₁) = \", dot(u1, v1), \", (u₁, v₂) = \", dot(u1, v2))\n",
    "println(\"(u₂, v₁) = \", dot(u2, v1), \", (u₂, v₂) = \", dot(u2, v2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Rayleigh's quotient\n",
    "\n",
    "**Definition 4 (Rayleigh's quotient)**: For any self-adjoint operator $\\mathbb{M}$ the quantity:\n",
    "\n",
    "\\begin{align}\n",
    "R(\\mathbb{M}, \\boldsymbol{\\phi}) = \\frac{(\\boldsymbol{\\phi}, \\mathbb{M} \\boldsymbol{\\phi})}{(\\boldsymbol{\\phi}, \\boldsymbol{\\phi})}, \\tag{9}\n",
    "\\end{align}\n",
    "\n",
    "is called *Rayleigh's quotient*.\n",
    "\n",
    "**Properties of Rayleigh's quotient**:\n",
    "\n",
    "Rayleigh's quotient is a function that maps vectors $\\boldsymbol{\\phi}$ to scalars. It satisfies the following properties:\n",
    "\n",
    "1. $R(\\mathbb{M}, \\boldsymbol{\\phi})$ is real for *any* $\\boldsymbol{\\phi}$ in the vector space.\n",
    "2. $R(\\mathbb{M}, c\\boldsymbol{\\phi}) = R(\\boldsymbol{\\phi})$.\n",
    "3. $\\lambda_{\\min}\\le R(\\mathbb{M}, \\boldsymbol{\\phi})\\le \\lambda_{\\max}$, where is $\\lambda_{\\min},\\lambda_{\\max}$ are the minimum and maximum eigenvalues of $\\mathbb{M}$.\n",
    "4. The stationary values of $R$ are the eigenvalues of $\\mathbb{M}$ and they are obtained when $\\boldsymbol{\\phi}$ is the corresponding eigenvector of $\\mathbb{M}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Exercise**: Show property 1 of Rayleigh's quotient.\n",
    "\n",
    "**Exercise** (bit harder): Show property 4 of Rayleigh's quotient. That is, assume that $R(\\mathbb{M}, \\boldsymbol{\\phi})$ is a stationary point and show that $\\boldsymbol{\\phi}$ is necessarily an eigenvector of $\\mathbb{M}$. \n",
    "\n",
    "Hint: To do so, write consider the Rayleigh's quotient evaluated at the slightly perturbed state $R(\\mathbb{M}, \\boldsymbol{\\phi}+\\delta\\boldsymbol{\\phi})$ with $\\|\\delta\\boldsymbol{\\phi}\\|\\ll\\|\\boldsymbol{\\phi}\\|$. Since $R(\\mathbb{M}, \\boldsymbol{\\phi})$ is a stationary point then it must be that\n",
    "\n",
    "\\begin{align*}\n",
    "R(\\mathbb{M}, \\boldsymbol{\\phi}+\\delta\\boldsymbol{\\phi}) - R(\\mathbb{M}, \\boldsymbol{\\phi}) = \\mathcal{O}\\big[(\\delta\\boldsymbol{\\phi})^2\\big].\n",
    "\\end{align*}\n",
    "\n",
    "From properties 3 and 4 it is evident that the maximum (or minimum) value of Rayleigh's quotient $R$ is obtained for the  eigenvector of $\\mathbb{M}$ that corresponds to $\\lambda_{\\max}$ (or $\\lambda_{\\min}$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Now we have all tools available to attack the question of maximum energy growth for linear system (1).\n",
    "\n",
    "### Perturbation energy growth\n",
    "\n",
    "The energy growth for linear system (1) is $E(t)\\big/E(0)$. Using solution (4)-(5) we get\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{E(t)}{E(0)} &= \\frac{\\big(e^{\\mathbb{A}t}\\boldsymbol{\\phi}_0\\,,\\,e^{\\mathbb{A}t}\\boldsymbol{\\phi}_0\\big)}{\\big(\\boldsymbol{\\phi}_0\\,,\\,\\boldsymbol{\\phi}_0\\big)}\\\\\n",
    "&= \\frac{\\big(\\boldsymbol{\\phi}_0\\,,\\,(e^{\\mathbb{A}t})^\\dagger e^{\\mathbb{A}t}\\boldsymbol{\\phi}_0\\big)}{\\big(\\boldsymbol{\\phi}_0\\,,\\,\\boldsymbol{\\phi}_0\\big)}\\\\\n",
    "&= \\frac{\\big(\\boldsymbol{\\phi}_0\\,,\\,e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}\\boldsymbol{\\phi}_0\\big)}{\\big(\\boldsymbol{\\phi}_0\\,,\\,\\boldsymbol{\\phi}_0\\big)}.\\tag{12}\n",
    "\\end{align*}\n",
    "\n",
    "But that's nothing else than the Rayleigh quotient of operator $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$!\n",
    "\n",
    "**Exercise**: Show that operator $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$ is self-adjoint and, furthermore, that its eigenvalues are positive.\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Maximum energy growth\n",
    "\n",
    "**Theorem IV (Bounds on perturbation energy growth)**\n",
    "\n",
    "The energy growth $E(t)\\big/E(0)$ of linear system (1) is bounded from above by the maximum eigenvalue of $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$. The inital state $\\boldsymbol{\\phi}_0$ that produces maximum energy growth at time $t$ is the eigenvector of $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$ that corresponds to its maximum eigenvalue.\n",
    "\n",
    "Similarly, $E(t)\\big/E(0)$ is bounded from below by the minimum eigenvalue of $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$ with the corresponding eigenvector being the initial condition that produces this minimum energy growth.\n",
    "\n",
    "*Proof*: This follows after the remark that $E(t)\\big/E(0) = R\\big(e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}\\,,\\, \\boldsymbol{\\phi}_0\\big)$ and then employing the properties of the Rayleigh quotient."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Corrolary 1**\n",
    "\n",
    "For $t\\ll1$, all eigenvectors of $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$ coincide with the eigenvectors of $\\mathbb{A^\\dagger} + \\mathbb{A}$.\n",
    "\n",
    "**Corrolary 2**\n",
    "\n",
    "For $t \\to \\infty$, the eigenvector of $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$ that corresponds to the largest eigenvalue coincides with the eigenvector of $\\mathbb{A^\\dagger}$ that corresponds to the eigenvalue with maximum real part.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Back to the initial motivation\n",
    "\n",
    "We are now in a position to answer some of the motivational questions we've posed.\n",
    "    \n",
    "What is the best way to initiate linear system (1) so that we get maximum energy for $t\\to\\infty$?\n",
    "\n",
    "- The answer to this comes from Corrolary 2 and it's the eigenvector of **the adjoint operator**   $\\mathbb{A}^\\dagger$ that corresponds to the eigenvalue with maximum real part.\n",
    "\n",
    "If we are interested in maximum energy growth for very short times $t\\ll 1$ do we have to initiate the linear system differently?\n",
    "\n",
    "- Yes! From Corrolary 1, the most energy growth for short times comes about if we initiate our linear system with the eigenvector of operator $\\mathbb{A} + \\mathbb{A}^\\dagger$ that corresponds to the maximum eigenvalue."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Remarks\n",
    "\n",
    "The theorem on bounds on perturbation energy growth is general and holds for both normal and non-normal operators.\n",
    "\n",
    "For normal operators we have that operators: $\\mathbb{A}$, $\\mathbb{A}$, $(\\mathbb{A}+\\mathbb{A}^\\dagger)$, and $e^{\\mathbb{A}^\\dagger t}e^{\\mathbb{A}t}$ *all* commute with each other and thus share the same eigenvectors. Therefore one needs not need to distinguish between them; only computing the eigenvectors of $\\mathbb{A}$ is enough to give you everything you need. Thus, for normal operators the notion of perturbation growth is *equivalent* with the notion of modal instability. However, this is not the case at all for non-normal operators. The theorem above generalizes the notion of modal instability to take into account non-normality and transient energy growth effects. It lies in the heart of the, so called, Generalized Stability Theory.\n",
    "\n",
    "Before proving Collolaries 1 & 2 let's see all these in action.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Example**\n",
    "\n",
    "Let's take the non-normal operator\n",
    "\n",
    "\\begin{align*}\n",
    "\\mathbb{A} = \\begin{pmatrix} -1 & 5 \\\\ 0 & -2\\end{pmatrix}\n",
    "\\end{align*}\n",
    "\n",
    "and compute $E(t)/E(0)$ with the following initial conditions:\n",
    "\n",
    "1. eigenvector of $\\mathbb{A}$ that has eigenvalue with maximum real part,\n",
    "2. eigenvector of $\\mathbb{A}^\\dagger$ that has eigenvalue with maximum real part, $\\mathbb{A}^\\dagger$,\n",
    "3. eigenvector of $\\mathbb{A}+\\mathbb{A}^\\dagger$ that has the maximum eigenvalue, and \n",
    "4. eigenvector of $e^{\\mathbb{A}^\\dagger t_*}\\,e^{\\mathbb{A} t_*}$ that has the maximum eigenvalue for some finite value of $t_*$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Example**\n",
    "\n",
    "1. The eigenvector of $\\mathbb{A}$ with eigenvalue with maximum real part is: $\\begin{pmatrix}1\\\\0\\end{pmatrix}$.\n",
    "2. eigenvector of $\\mathbb{A}^\\dagger$ with eigenvalue with maximum real part is: $\\frac1{\\sqrt{26}}\\begin{pmatrix}1\\\\5\\end{pmatrix}\\approx \\begin{pmatrix}0.196\\\\0.981\\end{pmatrix}$.\n",
    "3. eigenvector of $\\mathbb{A}+\\mathbb{A}^\\dagger$ that has the maximum eigenvalue is: $\\approx \\begin{pmatrix}0.773\\\\0.634\\end{pmatrix}$.\n",
    "\n",
    "We see that all of the above states are thoroughly different to one another. Let's see what's the evolution of the energy if we initiate our linear system with the above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "# A function that integrates (1) and computes E(t)\n",
    "\n",
    "function integrateforward(x0; tfinal=1)\n",
    "    t = range(0, stop=tfinal, length=1000)\n",
    "    energy = zeros(length(t))\n",
    "    upperbound = zeros(length(t))\n",
    "    dt = t[2]-t[1]\n",
    "    xt = x0\n",
    "    P = Matrix(I, 2, 2);\n",
    "    expAdt = exp(A*dt)\n",
    "    for j=1:length(t)\n",
    "        energy[j] = dot(xt, xt)\n",
    "        upperbound[j] = maximum(eigvals(copy(P')*P))\n",
    "        P = expAdt*P\n",
    "        xt = expAdt*xt\n",
    "    end\n",
    "    return t, energy, upperbound\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "Figure(PyObject <Figure size 1000x350 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = [-1 5; 0 -2]; \n",
    "S, U = eigen(A); u1 = U[:, 1]; u2 =U[:, 2]\n",
    "Sadj, V = eigen(copy(A')); v1 = V[:, 2]; v2 =V[:, 1]; #to make sure v1 <--> u1\n",
    "Sinst, Uinst = eigen(A + copy(A')); u01 = Uinst[:, 1]; u02 =Uinst[:, 2];\n",
    "topt = 0.3; Sopt, Uopt = eigen(exp(A'*topt)*exp(A*topt)); w1 = Uopt[:, 1]; w2 =Uopt[:, 2];\n",
    "tfinal = 2\n",
    "figure(figsize=(10, 3.5))\n",
    "t, energy, upperbound = integrateforward(u1; tfinal=tfinal)\n",
    "semilogy(t, upperbound, \"--k\", label=\"upper bound\")\n",
    "semilogy(t, energy, label=\"\\$ A\\$\")\n",
    "t, energy = integrateforward(v2; tfinal=tfinal)\n",
    "semilogy(t, energy, label=\"\\$A^\\\\dagger\\$\")\n",
    "t, energy = integrateforward(u02; tfinal=tfinal)\n",
    "semilogy(t, energy, label=\"\\$A +A^\\\\dagger\\$\")\n",
    "t, energy = integrateforward(w2; tfinal=tfinal)\n",
    "semilogy(t, energy, label=\"\\$e^{A^\\\\dagger t_*}e^{A t_*}\\$ for \\$t_*=0.3\\$\")\n",
    "xlabel(L\"t\"); ylabel(L\"E(t)/E(0)\"); legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "Figure(PyObject <Figure size 1000x350 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tfinal = 1\n",
    "figure(figsize=(10, 3.5))\n",
    "t, energy, upperbound = integrateforward(u1; tfinal=tfinal)\n",
    "semilogy(t, upperbound, \"--k\", label=\"upper bound\")\n",
    "semilogy(t, energy, label=\"\\$ A\\$\")\n",
    "t, energy = integrateforward(v2; tfinal=tfinal)\n",
    "semilogy(t, energy, label=\"\\$A^\\\\dagger\\$\")\n",
    "t, energy = integrateforward(u02; tfinal=tfinal)\n",
    "semilogy(t, energy, label=\"\\$A +A^\\\\dagger\\$\")\n",
    "t, energy = integrateforward(w2; tfinal=tfinal)\n",
    "semilogy(t, energy, label=\"\\$e^{A^\\\\dagger t_*}e^{A t_*}\\$ for \\$t_*=0.3\\$\")\n",
    "title(\"zooming in near \\$t=0\\$\")\n",
    "ylim(0.9, 2); xlabel(L\"t\"); ylabel(L\"E(t)/E(0)\"); legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "In this example we see that although $\\mathbb{A}$ has all its eigenvalues stable (i.e., with negative real part) still some initial conditions give energy growth by a factor of 2 before they decay. \n",
    "\n",
    "This goes back to the last motivational question: \"Can perturbations grow if the linear operator has no modal instabilities?\" Apparently the answer is: **yes**.\n",
    "\n",
    "For other non-normal operators this amplification factor can be huge. Consider for example the matrix:\n",
    "\n",
    "\\begin{align*}\n",
    "\\mathbb{A} = \\begin{pmatrix} -1 & Re \\\\ 0 & -2\\end{pmatrix},\n",
    "\\end{align*}\n",
    "\n",
    "with $Re=10^3$ or even higher..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "Figure(PyObject <Figure size 1000x350 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = [-1 1e3; 0 -2]; \n",
    "Sadj, V = eigen(copy(A')); v1 = V[:, 2]; v2 =V[:, 1]; #to make sure v1 <--> u1\n",
    "tfinal = 5\n",
    "figure(figsize=(10, 3.5))\n",
    "t, energy, upperbound = integrateforward(v2; tfinal=tfinal)\n",
    "semilogy(t, upperbound, \"--k\", label=\"upper bound\")\n",
    "semilogy(t, energy, label=\"\\$A^\\\\dagger\\$\")\n",
    "t, energy = integrateforward(u02; tfinal=tfinal)\n",
    "semilogy(t, energy, label=\"\\$A +A^\\\\dagger\\$\")\n",
    "semilogy(t, 1e6exp.(-2*t), \":k\", label=\"\\$\\\\propto e^{-2\\\\lambda_1 t}\\$\")\n",
    "xlabel(L\"t\"); ylabel(L\"E(t)/E(0)\"); legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We see that\n",
    "\n",
    "\\begin{align*}\n",
    "\\mathbb{A} = \\begin{pmatrix} -1 & 10^3 \\\\ 0 & -2\\end{pmatrix},\n",
    "\\end{align*}\n",
    "\n",
    "leads to transient perturbation energy growth by as much as a factor of $10^5$ at $t\\approx0.5$ before perturbation energy starts decaying at the rate of $2\\lambda_1=-2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We now prove the two Corrolaries.\n",
    "\n",
    "\n",
    "**Proof of Corrolary 1** (For $t\\ll1$, all eigenvectors of $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$ coincide with the eigenvectors of $\\mathbb{A^\\dagger} + \\mathbb{A}$.)\n",
    "\n",
    "This is so because\n",
    "\\begin{align*}\n",
    "\\frac{E(t)}{E(0)} &= \\frac{\\big(\\boldsymbol{\\phi}_0\\,,\\,e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}\\boldsymbol{\\phi}_0\\big)}{\\big(\\boldsymbol{\\phi}_0,\\boldsymbol{\\phi}_0\\big)}\\\\\n",
    "&\\approx \\frac{\\big(\\boldsymbol{\\phi}_0\\,,\\big[\\mathbb{I} + \\mathbb{A^\\dagger}t + \\mathcal{O}(t^2)\\big]\\big[\\mathbb{I} + \\mathbb{A}t + \\mathcal{O}(t^2)\\big]\\boldsymbol{\\phi}_0\\big)}{\\big(\\boldsymbol{\\phi}_0,\\boldsymbol{\\phi}_0\\big)} \\quad \\text{(for $t\\ll 1$)}\\\\\n",
    "&= 1 + t\\frac{\\big(\\boldsymbol{\\phi}_0\\,,(\\mathbb{A^\\dagger}+\\mathbb{A})\\boldsymbol{\\phi}_0\\big)}{\\big(\\boldsymbol{\\phi}_0,\\boldsymbol{\\phi}_0\\big)} + \\mathcal{O}(t^2),\\tag{13}\n",
    "\\end{align*}\n",
    "which is nothing else than the Rayleigh quotient of $\\mathbb{A^\\dagger}+\\mathbb{A}$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Proof of Corrolary 2** (For $t \\to \\infty$, the eigenvector of $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$ that corresponds to the largest eigenvalue coincides with the eigenvector of $\\mathbb{A^\\dagger}$ that corresponds to the eigenvalue with maximum real part.)\n",
    "\n",
    "This is a bit harder; we will prove it here for the case of finite-dimensional operators.\n",
    "\n",
    "First let's make a remark that about the eigenvectors of the adjoint $\\mathbb{A}^\\dagger$. If $\\mathbb{U}$ is the matrix whose columns are the eigenvectors $\\boldsymbol{u}_j$ of $\\mathbb{A}$ then we can get the matrix $\\widetilde{\\mathbb{V}}$ of the (non-unit-normalized) eigenvectors of the adjoint $\\mathbb{A}^\\dagger$ as:\n",
    "\n",
    "\\begin{align*}\n",
    "\\mathbb{\\widetilde{V}} = (\\mathbb{U}^{-1})^\\dagger.\\tag{14}\n",
    "\\end{align*}\n",
    "\n",
    "(The tilde is just to remind us that each of the column of $\\widetilde{\\mathbb{V}}$ is *not* normalized to have unit norm.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "To understand why (14) is true, remember the orthogonality property between eigenvectors of $\\mathbb{A}$ and eigenvectors of $\\mathbb{A}^\\dagger$ that correspond to non-complex-conjugate eigenvalues. Thus:\n",
    "\n",
    "\\begin{align*}\n",
    "\\underbrace{\\begin{pmatrix} \\begin{bmatrix} & \\boldsymbol{\\upsilon}_1^* & \\end{bmatrix} \\\\ \\vdots \\\\ \\begin{bmatrix} & \\boldsymbol{\\upsilon}_n^* & \\end{bmatrix}\\end{pmatrix}}_{=\\mathbb{V}^{\\dagger}}\\,\\underbrace{\\begin{pmatrix} \\begin{bmatrix} \\\\ \\boldsymbol{u}_1 \\\\ \\\\ \\end{bmatrix} & \\dotsb & \\begin{bmatrix} \\\\ \\boldsymbol{u}_n \\\\\\\\ \\end{bmatrix}\\end{pmatrix}}_{=\\mathbb{U}} = \\underbrace{\\begin{pmatrix} \\alpha_1 & 0 & 0\\\\0 & \\ddots &0 \\\\ 0 & 0 & \\alpha_n\\end{pmatrix}}_{=\\mathbb{D}\\text{ diagonal matrix}}\n",
    "\\end{align*}\n",
    "\n",
    "with $\\alpha_j \\equiv (\\boldsymbol{\\upsilon}_j,\\boldsymbol{u}_j)$ the inner product between the eigenvectors of $\\mathbb{A}$ and the eigenvectors of $\\mathbb{A}^\\dagger$ with conjugate eigenvalue. From (..) it follows that\n",
    "\n",
    "\\begin{align*}\n",
    "\\mathbb{I} &= \\underbrace{\\begin{pmatrix} \\frac1{\\alpha_1^*}\\begin{bmatrix} & \\boldsymbol{\\upsilon}_1^* & \\end{bmatrix} \\\\ \\vdots \\\\ \\frac1{\\alpha_n^*}\\begin{bmatrix} & \\boldsymbol{\\upsilon}_n^* & \\end{bmatrix}\\end{pmatrix}}_{=\\left(\\mathbb{V}\\mathbb{D}^{-1}\\right)^{\\dagger}}\\,\\underbrace{\\begin{pmatrix} \\begin{bmatrix} \\\\ \\boldsymbol{u}_1 \\\\ \\\\ \\end{bmatrix} & \\dotsb & \\begin{bmatrix} \\\\ \\boldsymbol{u}_n \\\\\\\\ \\end{bmatrix}\\end{pmatrix}}_{=\\mathbb{U}}\n",
    "\\end{align*}\n",
    "\n",
    "From the above we conclude that\n",
    "\n",
    "\\begin{align*}\n",
    "\\mathbb{V}= (\\mathbb{U}^{-1})^\\dagger \\mathbb{D} \\tag{15}\n",
    "\\end{align*}\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   V = Complex{Float64}[0.74 + 0.00im 0.73 + 0.00im; -0.55 - 0.38im 0.54 - 0.43im]\n",
      "U⁻¹D = Complex{Float64}[0.74 + 0.00im 0.73 - 0.00im; -0.55 - 0.38im 0.54 - 0.43im]\n"
     ]
    }
   ],
   "source": [
    "A = rand(2, 2) + im*rand(2,2); \n",
    "\n",
    "S, U = eigen(A); u1 = U[:, 1]; u2 =U[:, 2]\n",
    "Sadj, V = eigen(copy(A')); v1 = V[:, 1]; v2 =V[:, 2]\n",
    "\n",
    "D = Diagonal([dot(u1, v1), dot(u2, v2)])\n",
    "\n",
    "println(\"   V = \", V)\n",
    "println(\"U⁻¹D = \", inv(U)'*D)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "But the diagonal matrix $\\mathbb{D}$ is only a scale factor for each column of $\\mathbb{V}$. If we are not necessarily interested in unit-normed eigenvectors of $\\mathbb{A}^\\dagger$ then we just have (14).\n",
    "\n",
    "Having established (14) or (15) we can proceed to show that for $t\\to\\infty$ the eigenvector of $e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}$ with maximum eigenvalue tends to the eigenvector of $\\mathbb{A}^\\dagger$ corresponding to the eigenvalue with maximum real part.\n",
    "\n",
    "Consider the eigenanalysis of $e^{\\mathbb{A}t}$: \n",
    "\n",
    "\\begin{align*}\n",
    "e^{\\mathbb{A}t} &= \\mathbb{U} \\, \\underbrace{\\begin{pmatrix} e^{\\delta_1 t} & 0 & 0\\\\0 & \\ddots &0 \\\\ 0 & 0 & e^{\\delta_n t}\\end{pmatrix}}_{=\\mathbb{\\Delta}}\\, \\mathbb{U}^{-1}. \\tag{16}\n",
    "\\end{align*}\n",
    "\n",
    "**Exercise**: It's easy to show that $\\mathbb{A}$ and $e^{\\mathbb{A}t}$ commute and thus share the same eigenvectors. That is why in (16) the eigenvectors of $\\mathbb{A}$ appear. \n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "In eigenanalysis (16) the eigenvalues are ordered with descending real part, i.e., $\\mathrm{Re}(e^{\\delta_1 t})\\ge\\dotsb\\ge\\mathrm{Re}(e^{\\delta_n t})$. Also, $\\mathbb{U}^{-1}=\\widetilde{\\mathbb{V}}^{\\dagger}$ (from (14)). We can rewrite then (16) as\n",
    "\n",
    "\\begin{align*}\n",
    "e^{\\mathbb{A}t} = \\underbrace{\\begin{pmatrix} \\begin{bmatrix} \\\\ \\boldsymbol{u}_1 \\\\ \\\\ \\end{bmatrix} & \\dotsb & \\begin{bmatrix} \\\\ \\boldsymbol{u}_n \\\\\\\\ \\end{bmatrix}\\end{pmatrix}}_{=\\mathbb{U}}\\, \\underbrace{\\begin{pmatrix} e^{\\delta_1 t} & 0 & 0\\\\0 & \\ddots &0 \\\\ 0 & 0 & e^{\\delta_n t}\\end{pmatrix}}_{=\\mathbb{\\Delta}}\\,\\underbrace{\\begin{pmatrix} \\frac1{\\alpha_1^*}\\begin{bmatrix} & \\boldsymbol{\\upsilon}_1^* & \\end{bmatrix} \\\\ \\vdots \\\\ \\frac1{\\alpha_n^*}\\begin{bmatrix} & \\boldsymbol{\\upsilon}_n^* & \\end{bmatrix}\\end{pmatrix}}_{=\\widetilde{\\mathbb{V}}^{\\dagger}}.\\tag{17}\n",
    "\\end{align*}\n",
    "\n",
    "The eigenvalues are ordered with descending real part, i.e., $\\mathrm{Re}(e^{\\delta_1 t})\\ge\\dotsb\\ge\\mathrm{Re}(e^{\\delta_n t})$. Thus, for $t\\to\\infty$ the leading eigenvalue dominates and thus we have that \n",
    "\n",
    "\\begin{align*}\n",
    "e^{\\mathbb{A}t} \\sim \\frac{e^{\\delta_1 t} }{\\alpha_1^*}\\underbrace{\\begin{bmatrix} \\\\ \\boldsymbol{u}_1 \\\\ \\\\ \\end{bmatrix}  \\begin{bmatrix} & \\boldsymbol{\\upsilon}_1^* & \\end{bmatrix}}_{\\text{this is a matrix}}.\\tag{18}\n",
    "\\end{align*}\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Similarly,\n",
    "\n",
    "\\begin{align*}\n",
    "e^{\\mathbb{A}^\\dagger t} &= (\\mathbb{U}^{-1})^\\dagger \\,\\begin{pmatrix} e^{\\delta_1^* t} & 0 & 0\\\\0 & \\ddots &0 \\\\ 0 & 0 & e^{\\delta_n^* t}\\end{pmatrix}\\,\\mathbb{U}^\\dagger\\\\\n",
    "&= \\widetilde{\\mathbb{V}}\\,\\begin{pmatrix} e^{\\delta_1^* t} & 0 & 0\\\\0 & \\ddots &0 \\\\ 0 & 0 & e^{\\delta_n^* t}\\end{pmatrix}\\,\\mathbb{V}^{-1}.\\tag{19}\n",
    "\\end{align*}\n",
    "\n",
    "and for $t\\to \\infty$\n",
    "\n",
    "\\begin{align*}\n",
    "e^{\\mathbb{A}^\\dagger t} \\sim \\frac{e^{\\delta_1^* t}}{\\alpha_1} \\begin{bmatrix} \\\\ \\boldsymbol{\\upsilon}_1 \\\\ \\\\ \\end{bmatrix}  \\begin{bmatrix} & \\boldsymbol{u}_1^* & \\end{bmatrix}. \\tag{20}\n",
    "\\end{align*}\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Using (18) and (20) we have that\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{E(t)}{E(0)} &= \\frac{\\big(\\boldsymbol{\\phi}_0\\,,\\,e^{\\mathbb{A^\\dagger}t} e^{\\mathbb{A}t}\\boldsymbol{\\phi}_0\\big)}{\\big(\\boldsymbol{\\phi}_0,\\boldsymbol{\\phi}_0\\big)}\\\\\n",
    "&\\approx \\frac{e^{2\\mathrm{Re}(\\delta_1)t}}{\\left|\\alpha_1\\right|^2}\\frac{\\big(\\boldsymbol{\\phi}_0\\,,\\,\\boldsymbol{\\upsilon}_1\\,\\boldsymbol{u}_1^\\dagger \\boldsymbol{u}_1 \\boldsymbol{\\upsilon}_1^\\dagger\\boldsymbol{u}_0\\big)}{\\big(\\boldsymbol{\\phi}_0,\\boldsymbol{\\phi}_0\\big)}\\\\\n",
    "&= \\frac{e^{2\\mathrm{Re}(\\delta_1)t}}{\\left|\\alpha_1\\right|^2}\\frac{\\big(\\boldsymbol{\\phi}_0\\,,\\,\\boldsymbol{\\upsilon}_1\\,\\boldsymbol{\\upsilon}_1^\\dagger\\,\\boldsymbol{\\phi}_0\\big)}{\\big(\\boldsymbol{\\phi}_0,\\boldsymbol{\\phi}_0\\big)}\\quad\\text{(since }(\\boldsymbol{u}_1,\\boldsymbol{u}_1)=\\boldsymbol{u}_1^\\dagger \\boldsymbol{u}_1=1\\text{)}\\\\\n",
    "&= \\frac{e^{2\\mathrm{Re}(\\delta_1)t}}{\\left|\\alpha_1\\right|^2}\\frac{\\big(\\boldsymbol{\\upsilon}_1^\\dagger\\boldsymbol{\\phi}_0\\,,\\,\\boldsymbol{\\upsilon}_1^\\dagger\\,\\boldsymbol{\\phi}_0\\big)}{\\big(\\boldsymbol{\\phi}_0,\\boldsymbol{\\phi}_0\\big)}\\\\\n",
    "&= \\frac{e^{2\\mathrm{Re}(\\delta_1)t}}{\\left|\\alpha_1\\right|^2}\\frac{\\big|\\big(\\boldsymbol{\\upsilon}_1,\\boldsymbol{\\phi}_0\\big)\\big|^2}{\\big(\\boldsymbol{\\phi}_0,\\boldsymbol{\\phi}_0\\big)}, \\tag{21}\n",
    "\\end{align*}\n",
    "\n",
    "which is obviously maximized when $\\boldsymbol{\\phi}_0$ is parallel with $\\boldsymbol{\\upsilon}_1$. In that case\n",
    "\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{E(t)}{E(0)} &\\sim \\frac{e^{2\\mathrm{Re}(\\delta_1)t}}{\\left|(\\boldsymbol{\\upsilon}_1,\\boldsymbol{u}_1)\\right|^2}. \\tag{22}\n",
    "\\end{align*}\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "\n",
    "\n",
    "To understand what this means, consider equation (22) reduces to for the $2\\times2$ example. In that case we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{E(t)}{E(0)} &\\sim \\frac{e^{2\\mathrm{Re}(\\delta_1)t}}{\\sin^2\\theta}. \\tag{23}\n",
    "\\end{align*}\n",
    "\n",
    "where $\\theta$ is the angle between the two eigenvectors of $\\mathbb{A}$.\n",
    "\n",
    "Note that from (22) (or (23)) it is apparent that not only have we computed that the best initial condition for maximizing energy growth at $t\\to\\infty$ is the eigenvector $\\boldsymbol{\\upsilon}_1$ of the adjoint $\\mathbb{A}^\\dagger$, but we have also computed the growth that this initial condition would produce. We can see that the energy growth is larger by a factor of $1\\big/\\sin^2\\theta$ if we start of with $\\boldsymbol{\\upsilon}_1$ rather than just starting off with the eigenvector $\\boldsymbol{u}_1$of operator $\\mathbb{A}$ itself.\n",
    "\n",
    "Let's go back to our example with \n",
    "\n",
    "\\begin{align*}\n",
    "\\mathbb{A} = \\begin{pmatrix} -1 & 5 \\\\ 0 & -2\\end{pmatrix},\n",
    "\\end{align*}\n",
    "\n",
    "and see how well (23) is.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "Figure(PyObject <Figure size 1000x350 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = [-1 5; 0 -2]; \n",
    "S, U = eigen(A); u1 = U[:, 1]; u2 =U[:, 2]\n",
    "Sadj, V = eigen(copy(A')); v1 = V[:, 2]; v2 =V[:, 1]; #to make sure v1 <--> u1\n",
    "tfinal = 5\n",
    "figure(figsize=(10, 3.5))\n",
    "t, energy = integrateforward(u1; tfinal=tfinal)\n",
    "semilogy(t, energy, label=\"\\$ A\\$\")\n",
    "t, energy = integrateforward(v1; tfinal=tfinal)\n",
    "semilogy(t, energy, label=\"\\$A^\\\\dagger\\$\")\n",
    "semilogy(t, exp.(-2*t)/abs2(dot(u1, v1)), \"-.k\", label=\"eq. (23)\")\n",
    "xlabel(L\"t\"); ylabel(L\"E(t)/E(0)\"); legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Inner-product structure\n",
    "\n",
    "All the previous arguments were based on the fact that energy was given by the Euclidean inner product (2). In fact, in principle that's often note the case. Instead, the energy is given as:\n",
    "\n",
    "\\begin{align*}\n",
    "E(t) = \\big(\\boldsymbol{\\phi}\\,,\\,\\mathbb{M}\\boldsymbol{\\phi}\\big),\\tag{24}\n",
    "\\end{align*}\n",
    "\n",
    "where $\\mathbb{M}$ is a self-adjoint, positive-definite operatore. (An operator is positive-definite if and only if it's self-adjoint with positive eigenvalues.) The requirements on $\\mathbb{M}$ are to make sure that $E(t)>0$ and that it is only zero when $\\boldsymbol{\\phi}=0$.\n",
    "\n",
    "How do we find bounds on $E(t)/E(0)$ and also the initial conditions that give, e.g., maximum energy growth when $E(t)$ is given by (24)? Do we have to proceed through the whole analysis from scratch?  (No we don't!)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "It turns out that we can define a new variable $\\boldsymbol{\\psi}$ through:\n",
    "\n",
    "\\begin{align*}\n",
    "\\boldsymbol{\\psi} = \\mathbb{M}^{1/2}\\boldsymbol{\\phi}.\\tag{25}\n",
    "\\end{align*}\n",
    "\n",
    "(The square root of a positive-definite operator can be always defined.)\n",
    "\n",
    "In the new variable the energy is \n",
    "\n",
    "\\begin{align*}\n",
    "E(t) &= \\big(\\boldsymbol{\\phi}\\,,\\,\\mathbb{M}\\boldsymbol{\\phi}\\big)\\\\\n",
    "&= \\big((\\mathbb{M}^{-1/2}\\boldsymbol{\\psi})\\,,\\,\\mathbb{M}\\,(\\mathbb{M}^{-1/2}\\boldsymbol{\\psi})\\big)\\\\\n",
    "&= \\big(\\boldsymbol{\\psi}\\,,\\,(\\mathbb{M}^{-1/2})^{\\dagger}\\mathbb{M}\\,\\mathbb{M}^{-1/2}\\boldsymbol{\\psi}\\big),\\tag{24}\n",
    "\\end{align*}\n",
    "\n",
    "But $(\\mathbb{M}^{-1/2})^\\dagger = (\\mathbb{M}^\\dagger)^{-1/2} = \\mathbb{M}^{-1/2}$ and thus\n",
    "\n",
    "\\begin{align*}\n",
    "E(t) &= \\big(\\boldsymbol{\\psi}(t),\\boldsymbol{\\psi}(t)\\big).\\tag{25}\n",
    "\\end{align*}\n",
    "\n",
    "**Exercise**: Show that $(\\mathbb{M}^{-1/2})^\\dagger = (\\mathbb{M}^\\dagger)^{-1/2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "In variable $\\boldsymbol{\\psi}$ the energy has the form of (2). Now all we have to do is to write the linear system (1) for the new variable $\\boldsymbol{\\psi}$. Multiplying (1) from the left with $\\mathbb{M}^{1/2}$ we get:\n",
    "\n",
    "\\begin{align*}\n",
    "\\underbrace{\\mathbb{M}^{1/2}\\frac{\\mathrm{d} \\boldsymbol{\\phi}}{\\mathrm{d}t}}_{=\\frac{\\mathrm{d}\\boldsymbol{\\psi}}{\\mathrm{d}t}} &= \\mathbb{M}^{1/2}\\mathbb{A}\\!\\!\\!\\!\\underbrace{\\boldsymbol{\\phi}}_{=\\mathbb{M}^{-1/2}\\boldsymbol{\\psi}},\\tag{26}\n",
    "\\end{align*}\n",
    "\n",
    "which implies that \n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{\\mathrm{d} \\boldsymbol{\\psi}}{\\mathrm{d}t} &= \\underbrace{\\mathbb{M}^{1/2}\\mathbb{A}\\mathbb{M}^{-1/2}}_{\\equiv\\mathbb{A}_{\\mathbb{M}}} \\,\\boldsymbol{\\psi},\\quad \\boldsymbol{\\psi}(t_0)=\\boldsymbol{\\psi}_0. \\tag{27}\n",
    "\\end{align*}\n",
    "\n",
    "Therefore all we need to do is to do everything but with the only change $\\mathbb{A} \\to \\mathbb{A}_{\\mathbb{M}}=\\mathbb{M}^{1/2}\\mathbb{A}\\mathbb{M}^{-1/2}$. For example, the eigenmode of the $\\mathbb{A}_{\\mathbb{M}}^\\dagger$ with the eigenvalue iwth largest real part is the initial condition the gives maximum energy growth for $t\\to\\infty$ for system (27). To find the corresponding initial condition in $\\phi$ variables we just need to invert tranformation (25)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### References\n",
    "\n",
    "A good introduction for generalized stability theory is the review paper:\n",
    "\n",
    "- Farrell, B. F. and P. J. Ioannou (1996) \"Generalized stability theory. Part I: Autonomous operators.\" *J. Atmos. Sci.*, **53 (14)**, 2025-2040.\n",
    "\n",
    "\n",
    "An excellent linear algebra book that manages to make matrices and vectors \"come alive\" in front of you is that written by Strang:\n",
    "- Strang. Introduction to linear algebra. 5th edition, Wellesley-Cambridge Press, 2016\n",
    "\n",
    "His lectures at MIT are recorded and can be [watched online](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/).\n"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Julia 1.5.1",
   "language": "julia",
   "name": "julia-1.5"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}