1-numerical_methods/Topic2-Linear_Algebra.ipynb

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    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Matrix-vector-multiplication\" data-toc-modified-id=\"Matrix-vector-multiplication-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Matrix-vector multiplication</a></span><ul class=\"toc-item\"><li><span><a href=\"#Exercise:-Write-a-function-that-uses-for-loops-to-multiply-a-matrix-and-a-vector.\" data-toc-modified-id=\"Exercise:-Write-a-function-that-uses-for-loops-to-multiply-a-matrix-and-a-vector.-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>Exercise: Write a function that uses for loops to multiply a matrix and a vector.</a></span></li><li><span><a href=\"#Exercise:-Create-your-own-Vandermonde-matrix\" data-toc-modified-id=\"Exercise:-Create-your-own-Vandermonde-matrix-1.2\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>Exercise: Create your own Vandermonde matrix</a></span></li></ul></li><li><span><a href=\"#Norms,-inner-products,-and-orthogonality\" data-toc-modified-id=\"Norms,-inner-products,-and-orthogonality-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Norms, inner products, and orthogonality</a></span><ul class=\"toc-item\"><li><span><a href=\"#Exercise:-Create-an-orthonormal-version-of-the-Vandermonde-matrix\" data-toc-modified-id=\"Exercise:-Create-an-orthonormal-version-of-the-Vandermonde-matrix-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>Exercise: Create an orthonormal version of the Vandermonde matrix</a></span></li></ul></li><li><span><a href=\"#Rank,-Inverses,-and-Linear-Systems\" data-toc-modified-id=\"Rank,-Inverses,-and-Linear-Systems-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Rank, Inverses, and Linear Systems</a></span><ul class=\"toc-item\"><li><span><a href=\"#Exercise:-See-how-the-rank-changes-if-redundant-equations-are-selected.\" data-toc-modified-id=\"Exercise:-See-how-the-rank-changes-if-redundant-equations-are-selected.-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>Exercise: See how the rank changes if redundant equations are selected.</a></span></li></ul></li><li><span><a href=\"#Eigen-and-Singular-Value-Decompositions\" data-toc-modified-id=\"Eigen-and-Singular-Value-Decompositions-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Eigen and Singular Value Decompositions</a></span></li></ul></div>"
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   "source": [
    "# Linear Algebra\n",
    "\n",
    "Linear algebra is required for all engineers, but the conceptual aspects are often not taught or have been forgotten, so it is useful to have a refresher on some key concepts."
   ]
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    "## Matrix-vector multiplication\n",
    "\n",
    "First, some definitions:\n",
    "\n",
    "* Dot product or \"inner product\":\n",
    "\n",
    "$\\vec{a} \\cdot{} \\vec{b} = \\sum_i a_i b_i$\n",
    "\n",
    "* Matrix/vector multiplication:\n",
    "\n",
    "$\\bar{\\bar{A}} \\vec{x} = \\sum_j A_{ij} x_j = b_i$\n",
    "\n",
    "* Matrix/matrix multiplication:\n",
    "\n",
    "$\\bar{\\bar{A}} \\bar{\\bar{B}} = \\sum_j A_{ij} B_{jk}$\n",
    "\n"
   ]
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   "source": [
    "### Exercise: Write a function that uses for loops to multiply a matrix and a vector.\n",
    "\n",
    "Input: \n",
    "```\n",
    "A  = np.array([[0, 1], [2, 3]])\n",
    "B = np.array([0, 1])\n",
    "\n",
    "for i in range(A.shape[0]):\n",
    "    sum = 0\n",
    "    for j in range(A.shape[1]):\n",
    "        sum += A[i][j] * B[j]\n",
    "    print(sum)\n",
    "```\n",
    "Output:\n",
    "```\n",
    "1\n",
    "3\n",
    "```"
   ]
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   "source": [
    "We will explore matrix-vector multiplication conceptually by constructing a dataset from the \"Vandermonde\" matrix of polynomials and a weight vector, $\\vec{w}$ to construct a dataset of the form:\n",
    "\n",
    "$y_i = w_0 + w_1 x_i + w_2 x_i^2$\n",
    "\n",
    "it may be useful to write this with summation notation and compare it to the matrix-vector multiplication definition:\n",
    "\n",
    "$y_i = \\sum_{j=0}^2 w_j x_i^{\\: j}$\n",
    "\n",
    "First, we can use `numpy` to create a vector $x_i$:"
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9., 10.])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "%matplotlib inline \n",
    "import numpy as np\n",
    "import pylab as plt\n",
    "plt.style.use('../settings/plot_style.mplstyle')\n",
    "\n",
    "xi = np.linspace(0,10,11)\n",
    "xi"
   ]
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   "source": [
    "Now, we can create a new vector $z_i = x_i^2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  0.,   1.,   4.,   9.,  16.,  25.,  36.,  49.,  64.,  81., 100.])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
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   "source": [
    "zi = xi**2\n",
    "zi"
   ]
  },
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   "source": [
    "Next, let's create a vector that contains each of the weight parameters, $w_j$: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "wj = [1.5, 0.8, -0.2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now construct $y_i$ manually:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 2400x1800 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "yi = wj[0] + wj[1]*xi + wj[2]*zi\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.plot(xi, yi, '--o')\n",
    "ax.set_xlabel('$x_i$')\n",
    "ax.set_ylabel('$y_i$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This works, but we can create the same dataset using a Vandermonde matrix, which is a matrix of polynomials defined as:\n",
    "\n",
    "$X_{ij} = x_i^{\\: j}$\n",
    "\n",
    "In other words, each column of the matrix consists of a different polynomial.\n",
    "\n",
    "We can construct this matrix using `numpy`. First, we need to turn $x_i$ into a column vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_i vector shape: (11,)\n",
      "x_i column shape: (11, 1)\n"
     ]
    }
   ],
   "source": [
    "print(\"x_i vector shape: {}\".format(xi.shape))\n",
    "#xi_col = xi.reshape((xi.shape[0], 1)) #<- here we \"reshape\" the matrix into a column\n",
    "xi_col = xi.reshape(-1, 1) #<- this is equivalent, but less clear. It is a common shortcut.\n",
    "print(\"x_i column shape: {}\".format(xi_col.shape))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can \"stack\" these vectors together to create the Vandermonde matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  1.,   0.,   0.],\n",
       "       [  1.,   1.,   1.],\n",
       "       [  1.,   2.,   4.],\n",
       "       [  1.,   3.,   9.],\n",
       "       [  1.,   4.,  16.],\n",
       "       [  1.,   5.,  25.],\n",
       "       [  1.,   6.,  36.],\n",
       "       [  1.,   7.,  49.],\n",
       "       [  1.,   8.,  64.],\n",
       "       [  1.,   9.,  81.],\n",
       "       [  1.,  10., 100.]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi = xi_col\n",
    "X_vdm = np.hstack((xi**0, xi**1, xi**2))\n",
    "X_vdm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we can directly create $y_i$ using matrix-vector multiplication based on the definition of matrix-vector multiplication:\n",
    "\n",
    "$\\bar{\\bar{X}}\\vec{w} = \\sum_j X_{ij}w_j = \\sum_j x_i^{\\:j} w_j = w_0x_i^0 + w_1x_i^1 + w_2x_i^2+ ... w_nx_i^n = w_0 + w_1x_i + x_2x_i^2 + ... w_nx_i^n$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 2400x1800 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "yi_vdm = X_vdm@wj\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.plot(xi, yi_vdm, '--o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can verify that they are equal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yi == yi_vdm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sometimes it is also useful to use `isclose` instead of `==` since numerical methods are prone to very small errors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.isclose(yi, yi_vdm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `.all()` method is a good way to confirm that all values are equal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = np.isclose(yi, yi_vdm)\n",
    "M.all()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise: Create your own Vandermonde matrix\n",
    "\n",
    "Let's make a 2nd order Vandermonde matrix.\n",
    "\n",
    "Input:\n",
    "```\n",
    "x = np.linspace(0, 5, 6)\n",
    "vdm = np.hstack((x**0, x, x**2))\n",
    "print(vdm)\n",
    "```\n",
    "\n",
    "Output:\n",
    "```\n",
    "np.array([[1, 0, 0],\n",
    "                [1, 1, 1],\n",
    "                [1, 2, 4],\n",
    "                [1, 3, 9],\n",
    "                [1, 4, 16],\n",
    "                [1, 5, 25]])\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Norms, inner products, and orthogonality\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vectors can be described by various \"norms\" that capture their size and distance. The most common is the $L_2$ norm, also called the \"Euclidean distance\" which is defined as:\n",
    "\n",
    "$||\\vec{x}||_2 = \\sqrt{\\sum_i x_i^2}$\n",
    "\n",
    "This can also be computed by taking the square root of the vector-vector \"inner product\" of a vector with itself:\n",
    "\n",
    "$||\\vec{x}||_2 = \\sqrt{\\vec{x}^T \\vec{x}}$\n",
    "\n",
    "A vector is called \"normal\" if it's norm is 1. We can always \"normalize\" a vector by dividing it by its norm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Column norm: 3.3166247903554\n",
      "Normed column norm: 1.0\n"
     ]
    }
   ],
   "source": [
    "col_0 = X_vdm[:,0]\n",
    "norm_col_0 = np.linalg.norm(col_0,2)\n",
    "col_0_normed = col_0/norm_col_0\n",
    "print('Column norm: {}'.format(norm_col_0))\n",
    "print('Normed column norm: {}'.format(np.linalg.norm(col_0_normed,2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will also occasionally use the $L_1$ norm, defined as:\n",
    "\n",
    "$||\\vec{x}||_1 = \\sum_i |x_i|$\n",
    "\n",
    "These norms, and others are discussed in detail in the online notes of [Machine Learning Refined](https://github.com/jermwatt/machine_learning_refined/blob/gh-pages/notes/16_Linear_algebra/16_5_Norms.ipynb) and Lecture 3 of Trefethen & Bau."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is also useful to remember that the inner product between two different vectors is equal to the product of their magnitudes and the cosine of the angle between them:\n",
    "\n",
    "$\\vec{x}^T\\vec{y} = ||\\vec{x}||_2 ||\\vec{y}||_2 cos(\\theta)$\n",
    "\n",
    "This is described in detail in the online [Machine Learning Refined](https://github.com/jermwatt/machine_learning_refined/blob/gh-pages/notes/16_Linear_algebra/16_2_Vectors.ipynb) notes.\n",
    "\n",
    "We can use this to compute the angle between two normed vectors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "32.31153323742384"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "col_1 = X_vdm[:,1]\n",
    "norm_col_1 = np.linalg.norm(col_1,2)\n",
    "col_1_normed = col_1/norm_col_1\n",
    "\n",
    "cos_theta = np.dot(col_1_normed, col_0_normed)\n",
    "theta = np.degrees(np.arccos(cos_theta))\n",
    "theta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vectors are defined as \"orthogonal\" if their inner product is zero. We can check that this is consistent with our typical definition of \"orthogonal\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "90.0"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta = np.degrees(np.arccos(0))\n",
    "theta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In other words, orthogonal vectors are at right angles to each other, or have no projection onto each other."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One key concept that comes in handy is the ability to find the orthogonal components of an arbitrary set of vectors. We can do this by subtracting off the projection of one vector onto another:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-6.938893903907228e-17"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "col_1_ortho = col_1_normed - np.dot(col_0_normed, col_1_normed)*col_0_normed\n",
    "\n",
    "np.dot(col_1_ortho, col_0_normed)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "While this isn't technically zero, it is very close:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.isclose(np.dot(col_1_ortho, col_0_normed), 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a discussion and proof of why this works for orthnormal vectors in Lecture 2 of Trefethen & Bau.\n",
    "\n",
    "Let's take a look at the original vectors as compared to the orthonormal ones:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 3000x1800 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(2,1, figsize=(10,6))\n",
    "\n",
    "axes[0].plot(xi, col_0_normed, '--o')\n",
    "axes[0].plot(xi, col_1_normed, '--o')\n",
    "\n",
    "axes[1].plot(xi, col_0_normed, '--o')\n",
    "axes[1].plot(xi, col_1_ortho, '--o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that these vectors don't appear to be at \"right angles\" to each other. Remember that these are not 2-dimensional vectors, but rather 11-dimensional vectors. Our intuition only goes so far when working in high-dimensional spaces!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise: Create an orthonormal version of the Vandermonde matrix\n",
    "\n",
    "**Gram-Schmidt process for the first two columns**\n",
    "\n",
    "Input:\n",
    "```\n",
    "x = np.linspace(0, 5, 6)\n",
    "vdm = np.hstack((x**0, x**1, x**2))\n",
    "\n",
    "# Orthonormalizing the 1st column\n",
    "ortho_1 = vdm[:, 0] / np.linalg.norm(vdm[:, 0], order = 2)\n",
    "\n",
    "# Orthonormalizing the 2nd column\n",
    "ortho_2 = vdm[:, 1] - np.dot(vdm[:, 0], vdm[:, 1]) / np.dot(vdm[:, 1], vdm[:, 1]) * vdm[:, 1]\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rank, Inverses, and Linear Systems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The concept of the \"rank\" of a matrix is important. The formal definition of rank is the \"number of linearly independent columns/rows\". For an $m \\times n$ matrix, the rank is always less than or equal to the minimum of $m$ and $n$:\n",
    "\n",
    "rank $\\leq min(m, n)$\n",
    "\n",
    "It is sometimes convenient to think of rank in terms of a linear system of equations defined by:\n",
    "\n",
    "$\\bar{\\bar{A}}\\vec{x} = \\vec{b}$\n",
    "\n",
    "where $\\bar{\\bar{A}}$ is an $m \\times n$ matrix that defines equations in terms of unknown variables defined by $\\vec{x}$. If $m = n$ then $\\bar{\\bar{A}}$ is square and the number of equations is equal to the number of unknowns. As long as there are no redundant equations, then the rank of $\\bar{\\bar{A}}$ is equal to $m$ and $n$. However, if there are redundant equations then the rank is equal to the number of non-redundant equations and the system is **underconstrained**. On the other hand, if $m > n$ then there are more equations than unknowns, and the system is **overconstrained** (assuming there are no redundant equations).\n",
    "\n",
    "A matrix is **invertible** if and only if it is a square, full-rank matrix. This is equivalent to saying that a system of equations can only be solved ($\\vec{x} = \\bar{\\bar{A}}^{-1}\\vec{b}$) if the number of equations is equal to the number of unknowns (square matrix) and no equations are redundant (full-rank).\n",
    "\n",
    "Let's look at an example based on the Vandermonde matrix and our dataset from before. We will use 3 points to extract the weights, $w_j$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Shape of A: (3, 3)\n",
      "Rank of A: 3\n",
      "Weights: [ 1.5  0.8 -0.2]\n"
     ]
    }
   ],
   "source": [
    "A = X_vdm[1:4, :]\n",
    "b = yi[1:4]\n",
    "\n",
    "print('Shape of A: {}'.format(A.shape))\n",
    "print('Rank of A: {}'.format(np.linalg.matrix_rank(A)))\n",
    "\n",
    "A_inv = np.linalg.inv(A)\n",
    "w = A_inv@b\n",
    "\n",
    "print('Weights: {}'.format(w))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise: See how the rank changes if redundant equations are selected.\n",
    "\n",
    "Input:\n",
    "```\n",
    "A = np.array([[0, 1], [2, 3]])\n",
    "print(np.linalg.matrix_rank(A))\n",
    "\n",
    "# Let's add a redundant row here\n",
    "B = np.array([[0, 1], [2, 3], [2, 4]])\n",
    "print(np.linalg.matrix_rank(B))\n",
    "```\n",
    "\n",
    "Output:\n",
    "```\n",
    "2\n",
    "2\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In practice, it is very inefficient to solve systems of equations with matrix inverses. You should be familiar with [Gaussian Elimination](http://mathworld.wolfram.com/GaussianElimination.html) from your linear algebra course. There are many other ways to solve linear systems, such as the QR factorization or using eigenvalues. Unfortunately we don't have time to cover these methods in this course, and will simply solve systems using the `solve` function from `numpy`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_solve = np.linalg.solve(A,b)\n",
    "np.isclose(w_solve, w).all()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this course, solving systems of equations is easy as long as (1) you can write the system in the form $\\bar{\\bar{A}}\\vec{x} = \\vec{b}$, and (2) the matrix $\\bar{\\bar{A}}$ is invertible (i.e. the system can be solved)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Eigen and Singular Value Decompositions\n",
    "\n",
    "The eigenvalue problem for a matrix $\\bar{\\bar{A}}$:\n",
    "\n",
    "$\\bar{\\bar{A}} v_n = \\lambda_n v_n$\n",
    "\n",
    "where $v_n$ is the $n$th eigenvector and $\\lambda_n$ is the $n$th eigenvalue.\n",
    "\n",
    "To calculate eigenvalues of a matrix, use the `eigvals` function, and for calculating both eigenvalues and eigenvectors, use the function `eig`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues of A: [10.60311024  1.24543789  0.15145187]\n",
      "Eigenvectors of A: [[-0.13772903 -0.81480675  0.65820453]\n",
      " [-0.43070617 -0.49754289 -0.7324674 ]\n",
      " [-0.89192091  0.29755844  0.17394918]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ True,  True,  True])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from numpy.linalg import eigvals, eig\n",
    "\n",
    "print('Eigenvalues of A: {}'.format(eigvals(A)))\n",
    "\n",
    "vals, vecs = eig(A)\n",
    "print('Eigenvectors of A: {}'.format(vecs))\n",
    "\n",
    "np.isclose(vals, eigvals(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The eigenvectors of a **symmetric** matrix will always be orthonormal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_sym = (A.T + A)/2. #make A symmetric\n",
    "vals, vecs = eig(A_sym)\n",
    "\n",
    "vec0 = vecs[:,0]\n",
    "vec1 = vecs[:,1]\n",
    "vec2 = vecs[:,2]\n",
    "\n",
    "np.isclose(vec0@vec1,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Eigendecomposition is only possible for a square matrix. However, there is a similar concept called a \"singular value decomposition\", or SVD, that will work for any matrix:\n",
    "\n",
    "$A = \\hat{U}\\hat{\\Sigma}V^T$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ True,  True,  True])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from numpy.linalg import svd\n",
    "\n",
    "vecsL, vals, vecsR = svd(A_sym)\n",
    "np.isclose(vecsR[2,:], vecs[:,2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a square matrix, the SVD is equivalent to the eigendecomposition, although the order of the vectors will not always be the same. In general, eigenvalues and singular values are ordered from largest to smallest, but this is not guaranteed by the algorithms that compute them.\n",
    "\n",
    "The advantage of the SVD is that it will also work for non-square matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Original matrix shape: (11, 3)\n",
      "Left singular vectors shape: (11, 11)\n",
      "Right singular vectors shape: (3, 3)\n",
      "Singular Values: [160.3135455    5.20337928   1.22146387]\n"
     ]
    }
   ],
   "source": [
    "vecsL, vals, vecsR = svd(X_vdm)\n",
    "\n",
    "print('Original matrix shape: {}'.format(X_vdm.shape))\n",
    "print('Left singular vectors shape: {}'.format(vecsL.shape))\n",
    "print('Right singular vectors shape: {}'.format(vecsR.shape))\n",
    "print('Singular Values: {}'.format(vals))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a more detailed discussion of the SVD in Lecture 4 of Trefethen & Bau. We will not study it directly in this course, but it is an important part of some algorithms that we will use."
   ]
  }
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