{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![CC](https://i.creativecommons.org/l/by/4.0/88x31.png)\n",
    "\n",
    "This work is licensed under a [Creative Commons Attribution 4.0 International License](http://creativecommons.org/licenses/by/4.0/).\n",
    "\n",
    "# Linear algebra with PyTorch\n",
    "\n",
    "This notebook introduces some frequently used linear Algebra and tensor operations in PyTorch. For a general introduction to important concepts of linear Algebra, refer to:\n",
    "\n",
    "> [Introduction to linear Algebra](https://math.mit.edu/~gs/linearalgebra/) by Gilbert Strang\n",
    "\n",
    "The first version of this notebook was created by **Mahdi Maktabi** during his time as student assisant at TU Braunschweig's [Institute of Fluid Mechanics](https://www.tu-braunschweig.de/ism)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import PyTorch and plotting libraries\n",
    "import torch as pt\n",
    "import visualization as vis\n",
    "import matplotlib.pyplot as plt\n",
    "from math import sqrt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## PyTorch tensors\n",
    "\n",
    "PyTorch provides a [tensor](https://pytorch.org/docs/stable/tensors.html) type to define $n$-dimensional arrays. Each element in a tensor must be of the same type. Working with PyTorch tensors has a very similar look and feel to working with Numpy arrays. flowTorch uses PyTorch tensors as data structure for field data.\n",
    "\n",
    "### Creating tensors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# creating a vector of length 10 initialized with zeros\n",
    "origin = pt.zeros(10)\n",
    "origin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Dimension:  torch.Size([10])\n",
      "Datatype:  torch.float32\n"
     ]
    }
   ],
   "source": [
    "# the two most important tensor attributes: size and dtype\n",
    "print(\"Dimension: \", origin.size())\n",
    "print(\"Datatype: \", origin.dtype)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[1., 1.],\n",
       "        [1., 1.],\n",
       "        [1., 1.]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create a 2D tensor with 3 rows and two columns filled with ones\n",
    "ones = pt.ones((3, 2))\n",
    "ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([ 0.0000,  2.5000,  5.0000,  7.5000, 10.0000])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 1D tensor with 5 linearly spaced values between zero and ten\n",
    "x = pt.linspace(0, 10, 5)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Created tensor of shape  torch.Size([4, 3])\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "tensor([[ 0,  1,  2],\n",
       "        [ 3,  4,  5],\n",
       "        [ 6,  7,  8],\n",
       "        [ 9, 10, 11]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# creating a tensor from a Python list\n",
    "X_list = [\n",
    "    [0, 1, 2],\n",
    "    [3, 4, 5],\n",
    "    [6, 7, 8],\n",
    "    [9, 10, 11]\n",
    "]\n",
    "X = pt.tensor(X_list)\n",
    "print(\"Created tensor of shape \", X.size())\n",
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Accessing elements in a tensor\n",
    "\n",
    "PyTorch tensors support element access via the typical square-bracket syntax. Also slicing is supported."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(11)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# accessing the element in the fourth row and thrid column\n",
    "X[3, 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([0, 3, 6, 9])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# accessing the first column\n",
    "X[:, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([3, 4, 5])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# accessing the second row\n",
    "X[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([1, 4])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# accessing the first two elements in the second column\n",
    "X[:2, 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([7, 8])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# accessing the last two column of the thrid row\n",
    "X[2, -2:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basic tensor operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[2., 2., 2.],\n",
       "        [2., 2., 2.]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# elementwise addition of a scalar value\n",
    "# X += 2 is shorthand for X = X + 2\n",
    "X = pt.zeros((2, 3))\n",
    "X += 2\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[4., 4., 4.],\n",
       "        [4., 4., 4.]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# elementwise multiplication with a scalar value\n",
    "Y = X * 2\n",
    "Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[-2., -2., -2.],\n",
       "        [-2., -2., -2.]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# elementwise subtraction of two tensors with the same shape\n",
    "X - Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[8., 8., 8.],\n",
       "        [8., 8., 8.]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# elementwise multiplication of two tensors with the same shape\n",
    "X * Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Shape of 1D tensor:  torch.Size([1, 3])\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "tensor([[3., 3., 3.],\n",
       "        [3., 3., 3.]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# subtracting a 1D tensor from each row\n",
    "ones = pt.ones((1, Y.size()[1]))\n",
    "print(\"Shape of 1D tensor: \", ones.size())\n",
    "Y - ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Shape of 1D tensor:  torch.Size([2, 1])\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "tensor([[2., 2., 2.],\n",
       "        [2., 2., 2.]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# subtracting a 1D tensor from each column\n",
    "twos = pt.ones((Y.size()[0]), 1) * 2\n",
    "print(\"Shape of 1D tensor: \", twos.size())\n",
    "Y - twos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Shape of Y/Y.T:  torch.Size([2, 3]) / torch.Size([3, 2])\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "tensor([[4., 4.],\n",
       "        [4., 4.],\n",
       "        [4., 4.]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# transpose (swapping rows and columns in 2D)\n",
    "print(\"Shape of Y/Y.T: \", Y.size(), \"/\", Y.T.size())\n",
    "Y.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear algebra with PyTorch\n",
    "### Scalar product\n",
    "\n",
    "The scalar or dot product between two vectors $\\mathbf{a}$ and $\\mathbf{b}$ of length $N$ is defined as:\n",
    "\n",
    "$$\n",
    "  \\langle \\mathbf{a}, \\mathbf{b} \\rangle := \\sum\\limits_{i=1}^{N} a_i b_i.\n",
    "$$\n",
    "The dot product has couple of useful properties; see section 1.2 in [Introduction to linear algebra](https://math.mit.edu/~gs/linearalgebra/).\n",
    "\n",
    "The dot product of a vector with itself is equal to the squared length/magnitude\n",
    "$\\langle \\mathbf{a}, \\mathbf{a} \\rangle = \\left\\|\\mathbf{a}\\right\\|^2$\n",
    "\n",
    "Example:\n",
    "\n",
    "$\\mathbf{a} = [1, 2, 2]^T$, $\\langle \\mathbf{a}, \\mathbf{a} \\rangle = 1^2 + 2^2 + 2^2 = 9$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(9.)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = pt.tensor([1.0, 2.0, 2.0])\n",
    "len_a = pt.norm(a)\n",
    "\n",
    "assert pt.dot(a, a) == len_a**2\n",
    "pt.dot(a, a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cosine of the angle enclosed by two vectors of unit length is equal to their dot-product: \n",
    "$$\n",
    "  {\\displaystyle \\varphi =\\arccos{\\frac {\\langle \\mathbf{a} ,{\\mathbf {b}\\rangle}}{\\left\\|{\\mathbf {a}}\\right\\|\\left\\|{\\mathbf {b}}\\right\\|}}}\n",
    "$$\n",
    "\n",
    "Example:\n",
    "\n",
    "$\\mathbf {a} = [1,1,1]^T,\\ \\mathbf {b} = [4,2,2]^T$\n",
    "\n",
    "$\\left\\|{\\mathbf {a}}\\right\\| = {\\sqrt {{{1}^2}+{{1}^2}+{{1}^2}}} = 1.7321$,\n",
    "\n",
    "$\\left\\|{\\mathbf {b}}\\right\\| = {\\sqrt {{{4}^2}+{{2}^2}+{{2}^2}}} = 4.8990$,\n",
    "\n",
    "$\\langle \\mathbf{a}, \\mathbf{b} \\rangle = 1\\cdot 4+1\\cdot 2+1\\cdot 2 = 8$\n",
    "\n",
    "${\\displaystyle \\varphi =\\arccos {\\frac {8}{1.7321\\cdot 4.8990}=0.3398}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(0.3398)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = pt.tensor([1.0, 1.0, 1.0])\n",
    "b = pt.tensor([4.0, 2.0, 2.0])\n",
    "c = pt.dot(a, b)/(pt.norm(a)*pt.norm(b)) \n",
    "angle = pt.acos(c) # angle in radians\n",
    "\n",
    "assert pt.isclose(angle, pt.tensor(0.3398), atol=1.0E-3)\n",
    "angle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To find a unit vector with the same direction as a given vector, we divide by the vector's magnitude:\n",
    "\n",
    "$$ {\\displaystyle \\mathbf{u} = {\\frac { \\mathbf{a} }{\\left\\|{\\mathbf {a}}\\right\\|}}}$$\n",
    "\n",
    "Example:\n",
    "\n",
    "$\\mathbf{a} = [3,6,1]^T$, $\\left\\|{\\mathbf{a}}\\right\\| = \\sqrt {46}$\n",
    "\n",
    "$\\left\\|{\\mathbf {u}}\\right\\|$ = ${\\sqrt {(\\frac {3}{\\sqrt {46}})^{2}+(\\frac {6}{\\sqrt {46} })^{2}+(\\frac {1}{\\sqrt {46} })^{2}}}=1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(1.0000)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# using the div-function is equivalent to elementwise division of a tensor by a scalar in this example\n",
    "# e.g., b = a / pt.norm(a) yields the same result\n",
    "a = pt.tensor([3.0, 6.0, 1.0])\n",
    "b = pt.div(a, pt.norm(a))\n",
    "a_length = pt.norm(b)\n",
    "\n",
    "assert pt.isclose(a_length, pt.tensor(1.0), atol=1.0E-3)\n",
    "a_length"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Two vectors are orthogonal if their dot product is zero.\n",
    "\n",
    "Example:\n",
    "\n",
    "$\\mathbf{a}=[3, 0]^T$, $\\mathbf{b}=[0, 2]^T$\n",
    "\n",
    "${\\displaystyle \\varphi =\\arccos {\\frac {{3\\cdot 0 + 0\\cdot 2}}{\\sqrt {2}\\cdot{\\sqrt {3}}}=90^{\\circ} }}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(90.)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = pt.tensor([3.0, 0.0])\n",
    "b = pt.tensor([0.0, 2.0])\n",
    "c = pt.dot(a, b)/(pt.norm(a)*pt.norm(b)) \n",
    "d = pt.acos(c)\n",
    "angle = pt.rad2deg(d)\n",
    "\n",
    "assert pt.isclose(angle, pt.tensor(90.), atol=1.0E-4)\n",
    "angle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cross product\n",
    "\n",
    "The cross product between two three-dimensional vectors $\\mathbf{a}$ and $\\mathbf{b}$ is defined as:\n",
    "$$\n",
    "\\mathbf{a}\\times\\mathbf{b} = \\sum\\limits_{i,j,k=1}^{3} a_i b_j \\epsilon_{ijk} \\mathbf{e}_k,\n",
    "$$\n",
    "\n",
    "where $\\epsilon_{ijk}$ is the [Levi-Civita symbol](https://en.wikipedia.org/wiki/Levi-Civita_symbol), and $\\mathbf{e}_k$ is the $k$-th unit vector.\n",
    "\n",
    "Example:\n",
    "\n",
    "$\\mathbf{a}=[1, 2, 3]^T$, $\\mathbf{b}=[3, 2, 1]^T$\n",
    "\n",
    "${\\mathbf{a}}\\times{\\mathbf{b}}=\\begin{bmatrix}1 \\\\ 2 \\\\ 3 \\end{bmatrix}\\times\\begin{bmatrix}3 \\\\ 2 \\\\ 1 \\end{bmatrix}=\\begin{bmatrix}2\\cdot1 - 3\\cdot2 \\\\3\\cdot3 - 1\\cdot1 \\\\1\\cdot2 - 2\\cdot3\\end{bmatrix}\\,=\\begin{bmatrix}-4 \\\\ 8 \\\\ -4\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([-4.,  8., -4.])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = pt.tensor([1.0, 2.0, 3.0])\n",
    "b = pt.tensor([3.0, 2.0, 1.0])\n",
    "c = pt.cross(a, b)\n",
    "\n",
    "assert pt.equal(c, pt.tensor([-4., 8., -4.]))\n",
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cross product may be used to compute the angle between two vectors:\n",
    "\n",
    "$${\\displaystyle \\varphi =\\arcsin {\\frac {\\left\\| \\mathbf{a} \\times \\mathbf {b}\\right\\|}{ \\left\\|{\\mathbf {a}}\\right\\| \\left\\|{\\mathbf {b}}\\right\\|}}}$$\n",
    "\n",
    "Example:\n",
    "\n",
    "$\\mathbf {a} = [1,-2,3]^T,\\mathbf{b} = [-4,5,6]^T$\n",
    "\n",
    "$\\left\\|{\\mathbf {a}}\\right\\|= {\\sqrt {{{1}^2}+{{-2}^2}+{{3}^2}}} = \\sqrt {14}$,\n",
    "$\\left\\|{\\mathbf {b}}\\right\\|= {\\sqrt {{{-4}^2}+{{5}^2}+{{6}^2}}} = \\sqrt {77}$\n",
    "\n",
    "${\\displaystyle \\left\\|\\mathbf {a\\times b}\\right\\| ={\\begin{vmatrix}  \\mathbf i& \\mathbf j& \\mathbf k \\\\1&-2&3\\\\-4&5&6\\\\\\end{vmatrix}}}$\n",
    "${\\displaystyle {\\begin{aligned} =(-27)\\mathbf {i} -(18)\\mathbf {j} +(-3)\\mathbf {k}={\\sqrt {{{(-27)}^2}+{{(-18)}^2}+{{(-3)}^2}}=3{\\sqrt{118}}}\\end{aligned}}}$\n",
    "\n",
    "${ \\varphi =\\arccos {\\frac {3\\sqrt{118}}{\\sqrt {14}\\sqrt {77}}=83.0023^\\circ}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(83.0023)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = pt.tensor([1.0, -2.0, 3.0])\n",
    "b = pt.tensor([-4.0, 5.0, 6.0])\n",
    "c = pt.norm(pt.cross(a, b))/(pt.norm(a)*pt.norm(b)) \n",
    "angle = pt.rad2deg(pt.asin(c))\n",
    "\n",
    "assert pt.isclose(angle, pt.tensor(83.0023), atol=1.0E-4)\n",
    "angle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The magnitude of the cross product yields the area of the parallelogram spanned by two vectors $\\mathbf {a}$ and $\\mathbf {b}$.\n",
    "\n",
    "$$ A = {\\displaystyle \\left\\|\\mathbf {a} \\times \\mathbf {b} \\right\\|=\\left\\|\\mathbf {a} \\right\\|\\left\\|\\mathbf {b} \\right\\| \\sin \\varphi .}$$\n",
    "\n",
    "Example:\n",
    "\n",
    "$\\mathbf {a} = [1,1,1]^T$, $\\mathbf {b} = [1,1,2]^T$\n",
    "\n",
    "$\\mathbf{c}=\\mathbf{a}\\times\\mathbf{b} = [1, -1, 0]^T$, $\\left\\|\\mathbf{c}\\right\\|=\\sqrt{2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(1.4142)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = pt.tensor([1.0, 1.0, 1.0])\n",
    "b = pt.tensor([1.0, 1.0, 2.0])\n",
    "c_mag = pt.norm(pt.cross(a, b))\n",
    "\n",
    "assert pt.isclose(c_mag, pt.tensor(1.4142), atol=1.0E-4)\n",
    "c_mag"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Outer product\n",
    "\n",
    "The outer product of two vectors is a matrix. If the two vectors have dimensions $m$ and $n$, their outer product is an $m\\times n$ matrix. Consider the vectors $\\mathbf{a}=[a_{1},a_{2},\\dots ,a_{m}]^T$ and $\\mathbf{b} =[b_{1},b_{2},\\dots ,b_{n}]^T$. Their outer product, denoted by $\\mathbf{a}\\otimes \\mathbf{b}$, is obtained by multiplying each element of $\\mathbf{a}$ with each element of $\\mathbf{b}$ as follows:\n",
    "\n",
    "$$\n",
    "\\mathbf{C} = \\mathbf {a} \\otimes \\mathbf {b} ={\\begin{bmatrix}a_{1}b_{1}&a_{1}b_{2}&\\dots &a_{1}b_{n}\\\\a_{2}b_{1}&a_{2}b_{2}&\\dots &a_{2}b_{n}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\a_{m}b_{1}&a_{m}b_{2}&\\dots &a_{m}b_{n}\\end{bmatrix}}\n",
    "$$\n",
    "Example:\n",
    "\n",
    "$\\begin{bmatrix}0 \\\\ 1 \\\\ 2 \\end{bmatrix}\\begin{bmatrix}1&2&3 \\end{bmatrix}=\\begin{bmatrix}0&0&0\\\\1&2&3 \\\\2&4&6\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 840x840 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = pt.tensor([[0, 1, 2]]).view(3, 1)\n",
    "b = pt.tensor([1, 2, 3]).view(1, 3)\n",
    "C = pt.mm(a, b)\n",
    "\n",
    "assert pt.equal(C, pt.tensor([[0, 0, 0],[1, 2, 3],[2, 4, 6]]))\n",
    "vis.plot_matrices_as_heatmap([a, b, C], [r\"$a$\", r\"$b$\", r\"$C$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrix multiplication"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The product of two matrices $\\mathbf{C} = \\mathbf{AB}$ creates a new matrix with the same number of rows as $\\mathbf{A}$ and the same number of columns as $\\mathbf{B}$. The number of columns of matrix $\\mathbf{A}$ must be equal to the number of row of matrix $\\mathbf{B}$.\n",
    "\n",
    "$$\n",
    "{\\begin{bmatrix}c_{11}&c_{12}&\\cdots &c_{1p}\\\\c_{21}&c_{22}&\\cdots &c_{2p}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\c_{m1}&c_{m2}&\\cdots &c_{mp}\\\\\\end{bmatrix}}= {\\begin{bmatrix}a_{11}&a_{12}&\\cdots &a_{1n}\\\\a_{21}&a_{22}&\\cdots &a_{2n}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\a_{m1}&a_{m2}&\\cdots &a_{mn}\\\\\\end{bmatrix}}{\\begin{bmatrix}b_{11}&b_{12}&\\cdots &b_{1p}\\\\b_{21}&b_{22}&\\cdots &b_{2p}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\b_{n1}&b_{n2}&\\cdots &b_{np}\\\\\\end{bmatrix}}\n",
    "$$\n",
    "The coeffients of the resulting matrix are computed as follows:\n",
    "$$\n",
    "c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\\cdots +a_{ik}b_{kj}=\\sum _{k=1}^{n}a_{ik}b_{kj}\n",
    "$$\n",
    "\n",
    "Example:\n",
    "\n",
    "$$\n",
    "{\\mathbf{A}}{\\mathbf{B}}=\\begin{bmatrix}0&1&2 \\\\ 3&4&5 \\\\ 6&7&8 \\end{bmatrix}\\begin{bmatrix}0&1&2 \\\\ 3&4&5 \\\\ 6&7&8 \\end{bmatrix}=\\begin{bmatrix}15&18&21\\\\42&54&66 \\\\69&90&111 \\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1080x1080 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = pt.tensor(\n",
    "    [[0, 1, 2],\n",
    "     [3, 4, 5],\n",
    "     [6, 7, 8]]\n",
    ")\n",
    "B = pt.tensor(\n",
    "    [[0, 1, 2],\n",
    "     [3, 4, 5],\n",
    "     [6, 7, 8]]\n",
    ")\n",
    "C = pt.mm(A, B)\n",
    "\n",
    "assert pt.equal(C, pt.tensor([[15, 18, 21],[42, 54, 66],[69, 90, 111]]))\n",
    "vis.plot_matrices_as_heatmap([A, B, C], [r\"$A$\", r\"$B$\", r\"$C$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example 2:\n",
    "\n",
    "${\\mathbf{A}}{\\mathbf{B}}=\\begin{bmatrix}0&1&2 \\\\ 3&4&5 \\\\ 6&7&8 \\end{bmatrix}\\begin{bmatrix}0 \\\\ 3 \\\\ 6 \\end{bmatrix}=\\begin{bmatrix}15\\\\14\\\\23 \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 600x600 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = pt.tensor(\n",
    "    [[0., 1., 2.],\n",
    "     [3., 4., 5.],\n",
    "     [6., 7., 8.]]\n",
    ")\n",
    "B = pt.tensor([[0., 3., 6.]]).view(3, 1)\n",
    "C = pt.mm(A, B)\n",
    "\n",
    "# instead of the mm function we can also use the @ opertaor\n",
    "assert pt.equal(C, A @ B)\n",
    "assert pt.equal(C, pt.tensor([15., 42., 69.]).view(3, 1))\n",
    "vis.plot_matrices_as_heatmap([A, B, C], [r\"$A$\", r\"$B$\", r\"$C$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Identity matrix\n",
    "\n",
    "The identity matrix is a square matrix with a value of one on the diagonal and zero elsewhere.\n",
    "\n",
    "$$\n",
    "I={\\begin{bmatrix}1&0&0&\\cdots &0\\\\0&1&0&\\cdots &0\\\\0&0&1&\\cdots &0\\\\\\vdots &\\vdots &\\vdots &\\ddots &\\vdots \\\\0&0&0&\\cdots &1\\end{bmatrix}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ones = pt.ones(3)\n",
    "identity = pt.diag(ones)\n",
    "\n",
    "assert pt.allclose(pt.eye(3), identity)\n",
    "vis.plot_matrices_as_heatmap([identity], [r\"$I$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transpose of a matrix\n",
    "\n",
    "The transpose of a matrix is obtained by swaping its rows and columns.\n",
    "\n",
    "$$\n",
    "\\mathbf{A}=\n",
    "\\begin{bmatrix}\n",
    "a_{11} & \\dots & a_{1n} \\\\\n",
    "\\vdots &       & \\vdots \\\\\n",
    "a_{m1} & \\dots & a_{mn}\n",
    "\\end{bmatrix},\\quad\n",
    "\\mathbf{A}^T=\n",
    "\\begin{bmatrix}\n",
    "a_{11} & \\dots & a_{m1} \\\\\n",
    "\\vdots &       & \\vdots \\\\\n",
    "a_{1n} & \\dots & a_{mn}\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Example:\n",
    "\n",
    "$\\begin{bmatrix}1&3&5\\\\2&4&6\\end{bmatrix}^T=\\begin{bmatrix}1&2\\\\3&4\\\\5&6\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 600x600 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = pt.tensor(\n",
    "    [[1, 3, 5],\n",
    "     [2, 4, 6]]\n",
    ")\n",
    "AT = pt.tensor(\n",
    "    [[1, 2],\n",
    "     [3, 4],\n",
    "     [5, 6]]\n",
    ")\n",
    "\n",
    "assert (A.T - AT).sum().item() == 0\n",
    "vis.plot_matrices_as_heatmap([A, A.T], [r\"$A$\", r\"$A^T$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symmetric matrix\n",
    "\n",
    "A symmetric matrix is a square matrix that is equal to its transpose: $\\mathbf{A} = \\mathbf{A}^T$. The matrix coefficients are symmetric with respect to the diagonal.\n",
    "\n",
    "Example:\n",
    "\n",
    "${\\begin{bmatrix}1&2&6\\\\2&3&4\\\\6&4&5\\end{bmatrix}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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kXQt1+O5dxogGooKPiBBfQ0Sk3Qt1+O6/7DAbCBhj1uPtdHgDzYRFREK+26EQeCH4wBjTjX27HWYA/x9QizczFhHptNpst4MxJhLvhdQHBh+ZeLsd/G11TRGR9iLUux2OYt+Sw3ggDi9wv8J7vYe9f/UmItKphXrNd++bY24BnmbfX7Rlh/g6IiLtWqjD90K813LYHuLzdggX/3IC48b2J617MpWVNaxdt52HHl7Brl0t33/0+yOYdekkjpv4A/z+CNat38E99y5vdMyIEb2ZdekkevdKobCwjKeeWcmixR+HY0gHtVmTx/HjUT+ga3wstXUBNmbn8edl77IpZ1eLxyTFRPOHGccxYfChWGt5+/Mt3LJwBSWVVfVtpgw7jCtPOIYeXZPILizi3lfe57WNX4VjSO1K7t3/oXz1Z6TP/Tlxwwc026Y2fw/5/1xMxWdbMJGRJBwznJTzT8RE7oumolc+pGjxu9QVl+HvkUrKzJOI/UG/cA2jzYR0zdda+y8Fb8ushfl3LuHU0+/lgoseBiy333JGq8fMunQSw4f34pLLHufMc+6npKSS2245A2O8+rTuSdxx6xksXbqeU069h/l3LeHiX07gh+MHtv2ADnIvr/ucs+77P8be9Hcm3vEP3v9yG/+48DR8e794zZh/9jRSEuKZetc/mfanx0hJiOeOM6fW1x/eK535Z03j3lfe46gb7+evr7zPnWefyNDMtHAMqd0oeftjbHVNq21sIEDunU/gS4ilz/3XkHn7ZVR+tpWC/yyvb1P64QZ2P/Uq3S47nb6PXkficUeQO/8Jagv2tPEI2l6b3HAzxpxmjFlgjHk3+FhgjDm1La7Vnjzy6Ft88WUetbUBysqqePLplQwYkEZCQvObP/z+CKZNPZzHHn+HvJ3FlJdXc/8Dr9OvbyrDhvYEYOrU4WRlF/LiojXU1gZYt34HS5d9wmkzRodzaAelrfmFFNfPWA111pKaGE9yXEyz7Xt0SWTC4EO5c8lb7CmvZE95JXcueYtJQ/qTkZwIwNlHHc47X2zl1Y1fURsI8OrGr3j3i62cM/bwMI3q4FdbUMTuZ16j28WnttquctM2qr/ZRcp5J+KLi8HfrStdzzqekhWrCQSDu/jVlSROPILYIf0wkZEkTxmHPz2Fkrfa/zO7UN9w8wP/Babj3WgrClYdA5xrjHkJON1aWxvK67ZXY47oR25uEaWlVc3W9+6VQkyMn8827XsRuOLiCnJyizhsQBqfbMhiQP80Nm1q/CJxn3+ew9QThrVp39uLHw3qx51nTyMpNoZAwPL4u/+jsKyi2baDM7pRVVPL57n59WWf5+ZTXVvL4B7dyCkqYXCPbixb/0Wj4zZk5TJl2GFtOo72wlrLroeep+tpE4lM7dJq2+ptOfi7H0JEUnx9WfShmdiqGmpyCojuk071thySJh3Z6Ljo/plUbc1pg96HV6jXfK8FTgYeBG7d+9KRxpgM4DpgFjAHuDXE18UYk9VKdXqor/d9jR7dh5nnj+eGm19osU1cXBRAk3AuLakkLt6bLcfHRZGVtbtRfUlJJfHx2koN8PbnWxh38wMkx0YzY/QQ8opKW2wbHxPdaG13r+KKKhKive9FfHRUg9m0p6iiioQYfb0Bil/9CCwkTR7zrW0DFVX49nsW4kuIBcBWVHptyqvwxcc2bhMfS+3Oxj/z7VGolx3OBV6w1s5q+Jq91toca+1s4EXg/BBfs90ZN7Y/N/3xNG6f9xKrVm1psV15eTVAk2WJhMQYysu8ACgrr25Sn5gYQ1lZ87Ppzqqoooon3v+Ym08/gUHpqc22KausIrGZEE2Kjaa0yvtelFVVk7Rfm+TYaEqbCe3OpiavgD0vrCD1V6ceUHtfbDSB8spGZYFS71mJifVC2RcXTWC/ZyqBsor6+vYs1OHbh8bvVry/V4NtQs5a27OlB5DbFtf8Lo6fNITr5p7Czbcu5N33vmi17fYdBVRW1jB4UEZ9WVJSLOlpyXz5VR4AX32dx6AG9QCDBqbzVbBe9vEZQ6TPR5/Urs3Wb8rZRbQ/koENwnlgeipRkZFs+sbbIbHpm10M69n45trQzDQ++6blHRSdReWmbdSVVJA99wG2Xnw7Wy++HYC8e55k18MvNmkf1SeDmp2F1JWU15dVbc7GRPvxZ6TUt6na3HinatXmbKL7Nv6Zb49CHb57gNb2gPRj3zpwp3PqjNFcecUU5l73HKtWtzzj3aumpo5ly9dz4QXH0r17ErGxUcy6dBLbtuWzYaO3yrJ8+Sf07pXCj08ZRWSkj+HDenLitMN5cdGath7OQe+8Y0aRkhAHQNf4WK6fMZmaujrWbGv+jVS+2VPCW5s2c/VJP6JLXAxd4mK4+qQf8canX5NT5G3te/qj9fxoUD8mD+lPpM/H5CH9OXZQP57+aH3YxnWwih83jN73/pae8y6vfwB0++UMDvnplCbtYwb3IapHKgX/Xkqgoora/D0UPvc6iROPwBfl/SFs0gljKXlzDRWbtmJrayl+dSU1OQUkThgV1rG1hVCv+b4GXG6MWW6tfb1hhTFmAt6ab8uLnB3cr6+YQm1tHfPvOKtR+e+vfYZPNnhh+vLi3/Lne5bx2hufAvD3B99g1qWT+MeDFxLlj2Dd+u3Mvf45rPWOzdtZzJzrnuXyyyYz69JJFBaW88g/3+Kdd1ufVXcGxxzWm19NHENcdBRlldV8kpXLLx59nvySMgAykhNZ/JuZXPL4i/xvqze7mvPMMq778SSWX30RAG9t2swtC1fUn3P9jlzmPLOM30z9IX865ySyC4v5/dNL2ZClZxq+6Ch8wbXxRuWJcUQkxFGxaSu58xbQ609XEpnaBePzkXb1eeT/czHbLpuP8UeQcMzhpJw7rf7YhHHDqCsqZdf9z1JXVIY/sxvp15xPZEqXMI6sbRi7939xKE5mTF/gI7xXN1sFbAxWDcF7S6F84Chr7baQXfTA+pWVmZmZefQPryU/Xy+o1tZSUxN59ilv1nPcHQ+TV9zyTS75/tKSElhx7cUAjF/4V3Ir9DPeltJjE9l2+Z1kZ2dnB5c1v5NQv6rZVmPMkcAdwCl4gQtQAjwJzNUfYYiIhDB8g3t8fwDsttaea4wxQLdg9S4byim2iEg7F8obbgZYg/eW8VjPzuBDwSsi0kDIwtdaW433Hm6BUJ1TRKSjCvVWs/8CpweXHEREpAWh3mr2D+DfwHJjzL3A10D5/o10001EOrtQh+96vLeON8DkVtrpHY1FpFMLdfjejBe+IiLSilDv870xlOcTEemo2uzdi0VEpGUKXxERBxS+IiIOKHxFRBxQ+IqIOKDwFRFxQOErIuJASMLXGJNijPmNMeYmY8zYBuUXGWPeMcZsMMY8YozpFYrriYi0d9/7jyyMMel471qRGSz6gzHmXKAL8HcgC4gFLgKmGmNGWWvzv+91RUTas1DMfK/BC9qzgHHAx8BteO/XNtFa29ta2w34GZABXB2Ca4qItGuhCN8TgUettc9Zaz8CrsV7l+JF1tq39zay1j4FPAdMD8E1RUTatVCEb29gQ4PP975p5qpm2n4I9A3BNUVE2rVQhG814G/weVXwY3NvoVoWomuKiLRroQjCb4CGuxhKgAvZNwNuqC+wKwTXFBFp10LxkpJrgKP3fhJ8L7d/tdB2SrC9iEinForwnQukfVsjY0x3vLXh50JwTRGRdu17h6+1dgew4wDa7cRbjhAR6fR080tExAGFr4iIAwpfEREHFL4iIg4ofEVEHFD4iog4oPAVEXFA4Ssi4oDCV0TEAYWviIgDCl8REQcUviIiDih8RUQcUPiKiDgQitfzPSgYY7JaqU4PW0dERA5AhwnfA/H0X5+CQK7rbnR8vnTgcgCqh1RQVVHutj8dXHVsRP2/Txn4CcU1BQ570/El+VO4LwTn6TDha63t2VJdcFacGcbuiIi0Smu+IiIOKHxFRBxQ+IqIOKDwFRFxQOErIuKAwldExAGFr4iIAwpfEREHFL4iIg4ofEVEHFD4iog4oPAVEXFA4Ssi4oDCV0TEAYWviIgDCl8REQcUviIiDih8RUQcUPiKiDig8BURcUDhKyLigMJXRMQBha+IiAMKXxERBxS+IiIOKHxFRBxQ+IqIOKDwFRFxQOErIuKAwldExAGFr4iIAwpfEREHFL4iIg4ofEVEHFD4iog4oPAVEXFA4Ssi4oDCV0TEAYWviIgDCl8REQciXXegM/pgdQXXz9vNqrWVREQYhgyM4u2Fmfh8pknbwj11XDF3F0teK8cYOGlyHPfd0Y0uyRH1bZ57qZQ/zitgW3YtfXtGcsucFH4yPSGcQ2oXcu/+D+WrPyN97s+JGz6g2Ta1+XvI/+diKj7bgomMJOGY4aScfyImct9/laJXPqRo8bvUFZfh75FKysyTiP1Bv3AN46D1zgOf8d4/NhEZve9nc8CEDE6dP6bZ9hXF1bxyxzq+fjsXjKH/sWlMvXYEMUlR9W02vZrNW/d9SnFOOck94pgwewiDjs9s87GEg8I3zD5YXcH0c3P4yy2pLFqQQVSU4X/rqzBNcxeA82fnUV1t+fKDPgD87LJcfv7rnbz4eAYAK9dUMnN2Hk/cl8aPp8azaHkZ58/Oo3dmJEeOjAnXsA56JW9/jK2uabWNDQTIvfMJovpm0Of+a6grqyDvrn9T8J/lpF4wHYDSDzew+6lXSf/decQM7EXxG6vJnf8Eve6+ksiULmEYycEt8/BDOP9fEw6o7eK5q6mrDnDpS1MAWPj7Vbx0/f84496jAchev5tFc1fz4zuO5LAJGXz5Vg6L5q7mvIw4MoZ2bbMxhIuWHcJszi0FXPTTJGaelURcnI/ISMPY0TGYZtJ3244alr5ezl03pJKaEkFqSgR33ZDK4uVlbM/yguShBUVMOy6O009OwO83nH5yAlMnxvHggqJwD+2gVVtQxO5nXqPbxae22q5y0zaqv9lFynkn4ouLwd+tK13POp6SFasJBIO7+NWVJE48gtgh/TCRkSRPGYc/PYWStz4Ow0g6jqJvyvn6nTwmXTWcuK7RxHWNZtJVw/nyzVyKcsoB+PjZLfQfn8bg4zOJ8PsYfHwmhx6TxppntjjufWh0mPA1xmS19ADSXfcPoLw8wPurK4mIgHEn7qDbkM2MmbKD/75U2mz7tRuriI42jBgaXV82Ymg0UVGwdmM1AOs2VnHUqOhGx40ZGc3aT6rbbiDtiLWWXQ89T9fTJhKZ2qXVttXbcvB3P4SIpPj6suhDM7FVNdTkFNS3ienf+GlvdP9MqrbmhLrr7VLepiL+MnEJ909bxsI5q9iTVdZ8u8/3EBHlI21Qcn1Z2qBkIvw+dn7uTRx2fl5ExvDGM9yMYV3J27SnzfofTlp2CKPdewIEArDgmRIWPZHBqGHRLFpexs8uy6VHeiZHHxnbqH1xSYDkxKa/H7skRVBcEtjXpsH6L0CXLhEUlwbabiDtSPGrH4GFpMnNrzs2FKiowhfXeKnGl+B9T2xFpdemvApffOPvky8+ltqdu0PU4/Zr8PGZHD6jD0kZsZTurOSNv2zgyUve4xfPTiIqrnHUVJXWEp3gb3KO6EQ/VaXes4yqslpiEhu3iUnyU1VW23aDCKMOM/O11vZs6QHkuu4fQGKCt7RwwdmJjBkZQ2Sk4SfTE5h4TCwLlzWdISQl+igqaRqie4rrSAqGclKij6Kiusb1e+pISugw39rvrCavgD0vrCD1V6ceUHtfbDSB8spGZYHSCgBMrBfKvrhoAmUVjduUVdTXd2bdDksiuUccxhgS02KZftNoSnZWkLW2oEnb6ITI+pBtqKqkpj6Uo+MjqSxp3KayuIbo+I4xZ+wYo2gnkpMi6N/X3+LNtf2NHBpNVZVl/adVHD7EW1pY/2kV1dUwcqh3R3jE0GhWra1qdNzqdVWMHB7V5HydTeWmbdSVVJA994FG5Xn3PEnC0cObrAFH9cmgZmchdSXlRCTGAVC1ORsT7cefkVLfpmpzNgnjR9QfV7U5m/gxQ9p2MO2QMcb7WbdN69IGdaGuOsDOL4roPtBbetj5RRF1NQG6B5ciug9KJmdDYaPjcjYWkja4Sxv3PDw0PQqz2Rcl86+nS1i7oYpAwLJoeRlvf1jJaSc13RrWp5efEyfHcfVN+eQX1JFfUMfVN+Vz8pQ4evf0Zge/Oj+ZpW+U88LLpdTUWF54uZRlK8q5ZGZyk/N1NvHjhtH73t/Sc97l9Q+Abr+cwSE/ndKkfczgPkT1SKXg30sJVFRRm7+HwudeJ3HiEfiivK930gljKXlzDRWbtmJrayl+dSU1OQUkThgV1rEdjD5bnkV5oTcRKCuo5OUb1xCfEkPmyEOatE3uEUf/Y9N4/e4NlBdWUV5Yxet3b2DAhHSSM7xffKPO7MfX7+bx+evfUFcT4PPXv2Hze3mMOrNjbOvTzDfMrry4C+UVAWZckMOeojoO6xfFkw+mMXZ0DNuzahg2YTtL/tODY8d564oL/pbGFdft4rCjtwEw/Xhvn+9e446I4V9/S+O62ws4d1YefXtGsuBvaYzRNjN80VH4ops+A/AlxhGREEfFpq3kzltArz9dSWRqF4zPR9rV55H/z8Vsu2w+xh9BwjGHk3LutPpjE8YNo66olF33P0tdURn+zG6kX3O+tpkBG5bsYPkd66ipqCMm0U+vI1I556HxRMf7Kcop5+HTXuPsvx9Dr9GpAJxy25G8csc6Hjz5FQD6/yidqdfue0aRefghnHLbkbz5140snLOK5B5xnHL7kfQY1v63mQEYa5t5TtDBGGOyMjMzM7ev6QeBg2L5t2PzpePr/g4A4xf+ldyKEscd6tjSYxN5b8aVAMz77GKKa5qusUroJPlTuO+EpWRnZ2cH7yl9J1p2EBFxQOErIuKAwldExAGFr4iIAwpfEREHFL4iIg4ofEVEHFD4iog4oPAVEXFA4Ssi4oDCV0TEAYWviIgDCl8REQcUviIiDih8RUQcUPiKiDig8BURcUDhKyLigMJXRMQBha+IiAMKXxERBxS+IiIOKHxFRBxQ+IqIOKDwFRFxQOErIuKAwldExAGFr4iIAwpfEREHFL4iIg4ofEVEHFD4iog4oPAVEXFA4Ssi4oDCV0TEAYWviIgDCl8REQcUviIiDih8RUQcUPiKiDhgrLWu+xASxpisVqozfT4fGWmRQF24utSJRYCvGwC7Kkup6yA/YwerCGPoFpMAQEltIdYGHPeoYzPGR+muSgKBQI21Nuo7n6eThG8PoBbYGabuhFJ68GOu0150Hvp6h1d7/Xp3B2qstfHf9QQdJnw7qr2/VKy1PV33pTPQ1zu8OvPXW2u+IiIOKHxFRBxQ+IqIOKDwFRFxQOErIuKAwldExAFtNRMRcUAzXxERBxS+IiIOKHxFRBxQ+IqIOKDwFRFxQOErIuKAwldExAGFr4iIAwpfEREHFL4iIg4ofEVEHFD4iog4oPAVEXFA4Ssi4oDCV0TEAYWviIgDCl/plIwxW40xb7ruh3ReCl9pU8aY3saYJ4wxnxpj9hhjyo0xm4wx9xhjMlz3T8SVSNcdkA4vDegNLAJ2ADXAMOBi4CxjzEhr7S6H/RNxQuErbcpauwqYsH+5MeYd4BlgJnB3uPsl4pqWHcSVbcGPXQ6ksTGmqzHmTmPMl8aYKmPMLmPM68aYE/ZrN8EY85oxpji4xLHKGPPTAzh/X2OMNcbc2Ezd48YYu1/Zm8F1437GmEXB6xUYY/5qjPEbY2KCSys5xpgKY8wyY0yv/c5xY/Cag4wx840x3xhjKo0xK40xx+zX1meMucoY84kxptQYU2SM+cwYc9+BfP3k4KOZr4SFMSYaSASigSHAvGDVywdwbArwPnAY8BRwb/A844DjgVeD7U4BXgC+Ae4CyoDzgP8zxvS01t4VwiEBxAOvA68B1wCTgSuAauAHQARwK9AH+A2wADiumfMsCPZ1HpAEXAW8ZIzpZ60tCrb5A3AT8ARwP97EqT8wLcRjknCx1uqhR5s/gJ8DtsFjC/CzAzz2oeAxM5up8wU/RuDNpncD6Q3qY4A1QBWQ0aB8K/Bmg8/7Bq9xYzPXeNz7r9Ko7M1g+1/vV74KCADP7Vd+V7D94AZlNwbLFgKmQfnpwfJLG5StAV52/X3UI3QPLTtIuCwHTgB+AtyBN9Pr8m0HGWN8wNnAemvtgv3rrbWB4D+PwLux94i1NrdBfSXemnIUMP37DaGJOuDB/creAwzw92bKwZu97+9vNpiwQSuCHwc0KCsChhpjhn/HvspBRssOEhbW2hwgJ/jpC8aY5cCbxphqa+0jrRzaDUgG1n3LJfoFP25spm5v2aEH2t8DlGOtrdqvbE/w47YWyg9p5jxbGn5ird1tjAFIaVA8F2+GvN4YswUvoF8CFjb4BSTtiGa+4oS19i0gC7jwQA9pw+582/kjWiiva+WYlurMd2lrrf0A75fHGcAy4IfA88D7xpjYVvohBymFr7gUC3T9lja78GaNI76l3ebgxyHN1A3Zr01zdgc/NtefUM+YvxNrbam19r/W2lnW2kF4s+GxeMsy0s4ofKVNGWPSWig/B+9p9crWjg8+pX4KGGGMObeZ8+ydHa4BtgMXGWO6NaiPBn6LtwNhSSvXKQHy2G83gjFmLHB0a30MB2NMajPFHwc/pjRTJwc5rflKW5tvjBkGvIK3wyAeb4vYGXhrwDcewDmuw9vG9YQx5kTgA7wbaEfhBe7vrbV1xpjZeFvNVhljHgbK8baajQauCa47t+Z+4GZjzEt466l9gV8C6/n2mXdb+8wY8wHwEd5Wup7AZXg3Ll9w2TH5bhS+0tb+i/dU/ny8m2cBvBC+B7jTWrvz204QvAE1Dm+v62nAWXhLEeuARxq0W2yMOR64HpiD9/O9ATjXWvt/B9DXeXg3xM7FC/u1eLszLsJ9+N6Nt1vj13h7gfPwbrrdZq1tbTlFDlKm8Q4XEREJB635iog4oPAVEXFA4Ssi4oDCV0TEAYWviIgDCl8REQcUviIiDih8RUQcUPiKiDig8BURcUDhKyLigMJXRMQBha+IiAMKXxERBxS+IiIOKHxFRBxQ+IqIOKDwFRFx4P8BJGGWWdYHVkwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = pt.tensor(\n",
    "    [[1, 2, 6],\n",
    "     [2, 3, 4],\n",
    "     [6, 4, 5]]\n",
    ")\n",
    "AT = pt.transpose(A, 0, 1)\n",
    "\n",
    "assert pt.equal(A, AT)\n",
    "vis.plot_matrices_as_heatmap([A], [r\"$A$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Orthogonal matrix\n",
    "\n",
    "A matrix is orthogonal if its column vectors form an orthogonal basis. The column vectors are orthogonal if:\n",
    "\n",
    "$$\n",
    "\\langle\\mathbf{q}_i, \\mathbf{q}_j\\rangle=\n",
    "\\begin{cases}\n",
    "  0 & \\text{if} ~i\\neq j \\\\\n",
    "  \\neq 0 & \\text{if}~i=j\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "Example:\n",
    "\n",
    "${\\mathbf{Q}}{\\mathbf{Q^{T}}}=\\begin{bmatrix}1&-2&2 \\\\ 2&-1&-2 \\\\ 2&2&1 \\end{bmatrix}\\begin{bmatrix} 1&2&2 \\\\ -2&-1&2 \\\\ 2&-2&1 \\end{bmatrix}=\\begin{bmatrix}9&0&0\\\\0&9&0 \\\\0&0&9 \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1080x1080 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Q = pt.tensor(\n",
    "    [[1.0, -2.0,  2.0],\n",
    "     [2.0, -1.0, -2.0],\n",
    "     [2.0,  2.0,  1.0]]\n",
    ")\n",
    "QT = pt.transpose(Q, 0, 1)\n",
    "O = pt.mm(Q, QT)\n",
    "\n",
    "assert pt.allclose(pt.eye(3)*9, O)\n",
    "vis.plot_matrices_as_heatmap([Q, QT, O], [r\"$Q$\", r\"$Q^T$\", r\"$O$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Orthonormal matrix\n",
    "\n",
    "The vectors $\\mathbf{q}_i$ are orthonormal if:\n",
    "\n",
    "$$\n",
    "\\langle\\mathbf{q}_i, \\mathbf{q}_j\\rangle={\\begin{cases}0&{\\text{if}}~i\\neq j\\quad\\text{(orthogonal vectors)} \\\\1&{\\text{if}~i=j\\quad\\text{(unit vectors: $|\\mathbf{q}_i|$ = 1)}}\\end{cases}}\n",
    "$$\n",
    "\n",
    "A matrix $\\mathbf{Q}$ with orthonormal column vectors satisfies the relation $\\mathbf{Q}^T\\mathbf{Q} = \\mathbf{Q}\\mathbf{Q}^T = \\mathbf{I}$.\n",
    "\n",
    "Example:\n",
    "\n",
    "${\\mathbf{Q}^T}{\\mathbf{Q}}=\\begin{bmatrix}{\\frac {1}{\\sqrt{2}}}&{\\frac {1}{\\sqrt{6}}}&{\\frac {1}{\\sqrt{3}}}\\\\{\\frac {-1}{\\sqrt{2}}}&{\\frac {1}{\\sqrt{6}}}&{\\frac {1}{\\sqrt{3}}}\\\\0&{\\frac {-2}{\\sqrt{6}}}&{\\frac {1}{\\sqrt{3}}}\\end{bmatrix}\\begin{bmatrix}{\\frac {1}{\\sqrt{2}}}&{\\frac {-1}{\\sqrt{2}}}&0\\\\{\\frac {1}{\\sqrt{6}}}&{\\frac {1}{\\sqrt{6}}}&{\\frac {-2}{\\sqrt{6}}}\\\\{\\frac {1}{\\sqrt{3}}}&{\\frac {1}{\\sqrt{3}}}&{\\frac {1}{\\sqrt{3}}}\\end{bmatrix}=\\begin{bmatrix}1&0&0\\\\0&1&0 \\\\0&0&1 \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1080x1080 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Q = pt.tensor(\n",
    "    [[1./sqrt(2), 1./sqrt(6), 1./sqrt(3)],\n",
    "     [-1./sqrt(2), 1./sqrt(6), 1./sqrt(3)],\n",
    "     [0./sqrt(2), -2./sqrt(6), 1./sqrt(3)]]\n",
    ")\n",
    "QT = pt.transpose(Q, 0, 1)\n",
    "I = pt.mm(QT, Q)\n",
    "\n",
    "assert pt.allclose(pt.eye(3), I, atol=1.0E-6)\n",
    "assert pt.allclose(pt.mm(Q, QT), pt.mm(QT, Q), atol=1.0E-6)\n",
    "vis.plot_matrices_as_heatmap([Q, QT, I], [r\"$Q$\", r\"$Q^T$\", r\"$I$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hermitian matrix\n",
    "\n",
    "A Hermitian matrix is a complex matrix that is equal to its conjugate transpose:\n",
    "\n",
    "$$ \\mathbf{A}=\\overline{\\mathbf{A}}^T $$\n",
    "\n",
    "Example:\n",
    "\n",
    "${\\mathbf{A} = \\begin{bmatrix}1&3-{\\mathrm  {j}}&4\\\\3+{\\mathrm  {j}}&-2&-6+{\\mathrm  {j}}\\\\4&-6-{\\mathrm  {j}}&5\\end{bmatrix},} \\quad {\\overline{\\mathbf{A}} = \\begin{bmatrix}1&3+{\\mathrm  {j}}&4\\\\3-{\\mathrm  {j}}&-2&-6-{\\mathrm  {j}}\\\\4&-6+{\\mathrm  {j}}&5\\end{bmatrix}}$\n",
    "\n",
    "${\\mathbf{A}=\\overline{\\mathbf{A}}^T= \\begin{bmatrix}1&3-{\\mathrm  {j}}&4\\\\3+{\\mathrm  {j}}&-2&-6+{\\mathrm  {j}}\\\\4&-6-{\\mathrm  {j}}&5\\end{bmatrix}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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S3Xs1t2vDoiji+ouWceZ5fRhUsuGRdO+/28DQYVn07pvRPG3XcTnU1UYs+O+adJ93Gth1XOuzWmP2yOG9d9o/YyMBpGpraVxZTvbwYc3TQkYG2aUlNCwoa9O/oWwhZGaSXTK0eVp2yVDIyEi3NfVpuTyA7OHDaChruzx1ntF7jmTOmx+SSqaap81+7QOGjh5MflEe+cX5DBk1kFmvftDcnkqmmPPmh4zea1Qcm9wteIxsvh4zRjyEsGADzd1+QG6ypp5EftvLYomCXJI19ev6rHfpLKMwr7ldH2/MYQO54uWjqVndwBuPlNFrUMeXIuurGskpbHsI5RZlUVfV2Nwnt6h1n9zizOZ2bdjDd1VABCec2vE4/bVqKlMUFrc+t1DUK/25uio9rKWmqm2fwl4JFs73+9iWulu9TtWla2gir/W9BIm8PFL1de32T+S2U6/z8kjV1a3r097y6touT50nvzifqlU1raZVlVc3teURQvp+kupV1a36VJZXU1DsvSQd8RjZfJ4R7yYy8nNI1bTzj7m6joz8nHV9qlv3SVbVNrdr4+X3yubg00fy9ytndHj2Oqcwk/p2AnVd5RpymwJ6TmEmdZWt+9RVNDa3q2NlH63hLzet4qLr+29U//yiBFUVqVbTKlenPxcUpn+45he27VO1OkV+YUDqSCI3XUNTtbWtpqdqa0nktBMmcnPaDQup2trm8JHIzWl/ee2EE6UdeeqhTK64s/k1YNjG1YaWaipqKOzd+mbvwj4FTW21VFekv5OC3gWt+hT1KWhuU1seI5uvxwTxKIpKO3oB3X5sRs6oQURrktTNXdI8rW7uEqLGZPNwlZxRg9rcmFk3Z1Gr4SzaeFEqItWYYvm86nbbh4wpprEhxaLZ64L6otkVJNdEzcNVhowpbnNj5oKZq1sNZ1H7pr9WR8WqJF87oYzj957L8XvPBeDyc5dwwyXL2vTfaddsFi1Yw+ryZPO0WdPryc0LlO6QHvu/027ZzJre+grRrLfr2Xk3f1ndlrpbvU7k5ZHZtw8NLW4Si5JJGsrKyC4tadM/u2QoNDbSsHBdPW5YuBCSyeZL8dklQ2mY3/qms4b588kuabs8pf3rnhc4ofiM5tey+cs3eRlz3prL6L1GkWjxhLKd9x3NwjmLqamspaaihkUfLmWX/UY3tycyEuzQNKRF7fMY2Xw9Joh3R1EyRaqhkagxHRyiNY3pz6m2T4fIHtibwn12ZMmfnqaxoobGihqW/OlpCvfbmawB6THIfY7Zh6rX36filVlEjUkqXplF1Rsf0Gf8Ptt0v7qrF++cS+XydEirWlnP5GvfISMrwYi92r+Zts/QPHb55AAev3EW1eUNVJc38PiNsxhz+EB6D0lfTtt/4jDe+/cyZj6zmOSaFDOfWcx7LyzjgFOGtbtMrXPEhALufX44f3istPkF8P3rBnDOD/u26b/H/rkM3yGb31y3gpqqFEvKGvnjL8uZcEoROTnpUvfZ04uZcn8l016tZU1DxMN3VbDgwzWMP7lom+6bup+iQw9m9bPP07BoEamGNax64ilCRgb5e+zepm9m377k7TaGlY88SrKqmmRVNSsfeZS8sbuR2SddTwoPPoiad2ZRPf1tomSS6ulvU/vuLIoOOXBb71q3lpGZQVZOFhmZGYQAWTlZZOVkddj/hb9PJZVM8aWrv0B2bjYjxw5j4oUnMPmWJ5v7/OO3TzLxwhMYOXYY2bnZfOnqL5Bck+SFh17dFrvUbXmMbB6vj8do9XPTWXjT5ObPs/73egBGXHMm2YP78MEFtzD88lMpGDsCgJLvfI5Fv3+cD75xE7DuOeJr5e9SSsl3PsfSO/9J2c8fJGtgb0q+87k2N3CqfR+8vJznbptDQ22S3IJMSnbvxVdu25/iAenLYKsW1fKrE/7Nl363L6P2SQfBidfvweTr3uHGY58HYMxh6eeIrzV8XB8mTtqDJ//vPe67aBq9S/I45fpxlH6i9zbfv+4mNy9Bbl7bcwW9+iQo7p3BtFdruejLi7njqWEMKskkkQhcf/sgfnHFcj53wEdkZQeOOr6Qb17Sr3neIyYUsnJ5kmu/u4zyFUlG7JjFT/8wmEFDLYXasOIjDidVV8/iW35PVFdH9rBSBp3zNRI5OTSWl1M26QYGnfNVckfvAED/005l5YMPseDaScC6ZySvlTtyBANO/yLlj05h2R13k9m3D/1PP5Wc4cPj2L1u63u//wafPuvw5s9Tau8B4OjERCA9nOU7vzuHE4rPAKC2qo5Lxl/LBTd/lQeX/5GailoevfUpHvzlo83L+NuNk8kvyuOnT19BfnEe7/1nDpccex111T1rbHJn8xjZPCGKNvxs3p4ghLCgpKSkZOc/fJcldT6tIm6Dcov51zEXAnDz7DOobFzxMXNoayrK7Mf5u9wJwNT5h9CQ7HIjA7Yr2RmD+fxBH1FWVlbWNFRju7K2Xg+7+nIWV1XFvTnbtcGFhbx89jkAfHHYOSwvWxnzFm3f+pf05a/zbwXgoD/c6vHRBQwuLGT+lddsUb12aIokSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFIPMuDegs4QQFmygefA22xBJ0gZZryUprccE8Y1x0si3qGxcEfdmbPeKMvs1//cj93yKJaurYtwaDepVyPlXp//7VwfvzvKyofFu0Hauf0lf4KO4NyN2Lx5/F6QWx70Z27fEYOAcAOZetTeLq6zVcaoqLGz+b4+PLiIxmOFXbtkiekwQj6KotKO2prMvJdtwcyRJHbBeS1KaY8QlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYZHbmwkIIeUBRFEVLW0zrD5wL9AP+GkXRK525TknSprNeS1L8OjWIA78F9gLGAYQQsoEXgZ2a2r8RQvhUFEVTO3m9kqRNY72WpJh19tCUQ4B/tPg8kXRR/yKwG/ARcHEnr1OStOms15IUs84O4oOBuS0+Hwe8GUXRfVEUzQJuB/br5HVKkjad9VqSYtbZQTxJ6+EuhwPPtvi8nPTYQ0lSvKzXkhSzzg7i7wPjAUIIh5A+4/LPFu3DgPJOXqckadNZryUpZp19s+YtwB9CCDOAEuBD4JkW7Z8E3u7kdUqSNp31WpJi1qlBPIqiP4UQksCJwKvAT6IoWgMQQugH9CFd/CVJMbJeS1L8OvuMOFEU3QHc0c70FcA+nb0+SdLmsV5LUrw6dYx4COHLIYQRnblMSVLns15LUvw6+4z4H4AohDCP9N33zwHPRlE0v5PXI0naMtZrSYpZZwfx/YEjST8G62TgLNKF/kPShf5Z4LkoihZ28nolSZvGei1JMevsmzX/A/wH+FkIIYN0oT+cdLH/IvAVIOrs9UqSNo31WpLi19nPEW8WRVESmAP8l/RjsZYBgfQfkZAkdRHWa0mKR6ee6Wh65NXhwBFNrzFACngduBf4F/BCZ65TkrTprNeSFL/OvuS4hPSlzGnA48BFwP+Loqiyk9cjSdoy1mtJillnB/G1Q11ygOymV0Ynr0OStOWs15IUs84O4utf6jwfSIUQppO+A/9feMZFkroC67Ukxayzn5pSDjzU9CKEMIB1d+GfCHwHaCR9BkaSFBPrtSTFb6s9NSWEkAnsBOzc9CohfRd+1tZapyRp01mvJSkenf3UlP1Zd5nzECCfdDH/ALiDdX+9TZIUI+u1JMWvs8eIv9L0/iFwH+v+MltZJ69HkrRlrNeSFLPODuJfBp6NomheJy+3R/rnLe/z5uQyqletISMzULJbL4753i4MHVPc4Ty1q9cw+SczmfX8MkKAXT41gBMuHUte8boryG8/tYinb3qfVQtr6TM0j6O/tTO7Hz14W+xSt9e3MI/vnfAp9hldQp/CPMqra3n8jdn89vFXWJPs+G+bDO5dxKUTj2Tf0aU0NCZ54s3Z3PDw8zQmU819vnDoOM46ch/6FuYzd8lKfvbQ87z+XzPPxvj0lw7n5O8dz+AdBlJXXc9z973ILd/+U4f9R31iOOffdDY77bMDNatreOy2Z7jz6r+16nPmVadw3FePIr9XPu+//l9uOu925s6cv7V3pSuxXm+Cex+u5Ld/Ws20d+qprIqonz+azMzQYf/yVUku+NEyHnumhhDguKPyuXnSAHr3WvdgmgcereKK61fwUVkjI0szuebifpw0oXBb7E63F0URq554iqqXp5KqqyW7tJR+E08ie8iQdvsna2pY+eBD1Mx8F4D8sbvS9/MnkZGf19yn+q1plE95gmR5OZl9+tJ7wrEUjPvENtmf7s7jY/N16hjxKIr+YlHfeHscO4Tz7juEK185moufPZIdD+7Pn77+Gqlk1OE89188jaoVDXz/8cO4cMphVK1o4IFLpze3z5++ir9dPJ1Pf2tnrnzlaI7+1s7cf/E0FsxYvS12qdvLy8lm7tKVfO2WBzno4t/w9Vse5FO7jeK7Jxza4TwhwM1fP5GKmjr+58rb+N+f380+o0u48IRPNfc5etxOfGvCIVx295Mccskt/H3qTH5zzmcZ1LvnFZXOdvL3PsPpV0zk1+fdxmd7f4kzdziPp/78XIf98wpzmfTEZcx8aTYnDzibi8dfx7FnH8VJ35nQ3Gfi90/gmC8fwcXjr+PkAWcz86XZTHriMnILcrfBHnUN1utN06dXgm+c1Ytf/HjARvU/4/wlLF2e5P2XR/DeSyNYujzJWd9e2tw+9Y06zjx/Cddc3I9V7+3Aj3/YjzPOX8J/3qrbWrvQo1Q8+xxVU19l0De+xrBrf0zuqFEs+d1tpOrr2+2//K57SFZWUXrZJZRedgnJyiqW33Nvc3v93I9Ydtdf6TPhWIZffx29J4xn+V13Uz9vu/rlfLN5fGy+rXKzZgjhcyGEO0IILzS97gghfHZrrKs7GzCqkLxeTWeyo4hEBlSvbKBmdUO7/csX1jL738s47vtjKOiTTUGfbI77/hjefXYpqxbVAjD1/nnsfOgAdj96MBlZCXY/ejA7HzKAqff583ZjlK1Yze3PvMaCFauJIpi/fDUPT53JfjsO63CevXcoYdSgvtzw8PNU1zewqLySm6e8xEkH7k52Zvq3+y8cOo6Hp87k9TllNCZT3PfCND5atooT9x+7rXatW8ovyuPMq77ALd/+IzNemEUqmaKupp4P3vyww3kOPekAEhkJ/nz5vTTUNTB3xjz+duNkTjxvfHOf4795DA/8/B/MnTGPhroG/nz5vWRmZ3Lo5/bfFrvVpVivN84xRxTwxc8VscOIj7+Q/NH8NTz+zxpuuLI//ftl0L9fBjdc2Z9/PFnNvAVrALj1jtWMPyKfz3+mkKyswOc/U8gxh+fzuzs8abIxKl94iV5HHE720CEksrPofdwxRI1JaqbPaNO3ceVKat+ZRd8TjyejsICMwgL6nng8tTNm0lhenl7eSy+Tv+sYCsbtQcjIoGDcHuSNGUPliy9v4z3rnjw+Nl+nBvEQQlYIYTLwAHA6MLbpdTrwYAjhkaa789Vk1vNL+fFBT3PF3k8x5WezOOTMkRT2bf9pYYtmVZCZnWBIi6ErQ8YUk5EVWDSrorlP6Sd6tZqvdPdeze3adAftMoJZZUs7bB9TMpCyFatZVb3uN/WZ85aQl5PFiAF9mvoMYMa8xa3mmzlvCWNKNu7swfZqt4N3Ia8wl9Kdh/Dn2b/m/sW3M+nxS9lhjxEdzjN6z5HMefNDUi2GBc1+7QOGjh5MflEe+cX5DBk1kFmvftDcnkqmmPPmh4zea9RW3Z+uxHq99bw1s56cnMC4setq+bixOWRnw1sz0ydaps2sZ/+9Wtf6/fbM4a232z8Ro3VStbU0riwne/i6EyQhI4Ps0hIaFrQd7tdQthAyM8kuGdo8LbtkKGRkpNua+rRcHkD28GE0lDl8sLN5fLTW2WfELwE+A9wKlEZR1CeKoj6kH4X1W+B44OJOXicAIYQFHb2ALjtAesxhA7ni5aO57MWjOPYHYxg+rneHfeurGskpbPtzMbcoi7qqxuY+uUWt++QWZza3a9N8/dMHMKZkIDc/9lKHfQpys6msbX05tKI2HcoLc7M32Kcw10c0b0iv/kUAHHzi/lx4xFWcPvJc5kyby08ev5T84vx258kvzqdqVU2raVXl1U1teRQUp8eEVq+qbtWnsry6uW07Yb3eSioqU/QqavvjtXdxBhWVqXV9erX+Q6a9e2dQUZVqM59aS9Wla2kir/XxmsjLI1XfduhCqq6eRG7bYWeJvDxSdXXr+rS3vLqeNxQibh4frXV2ED8NeCiKonOjKFq4dmIURYuiKDofeBg4o5PX2SPk98rm4NNH8vcrZ3R49jqnMJP6dgJ1XeUacpsCek5hJnWVrfvUVTQ2t6u14/YZwys/Pa/5Nbh3UXPbeccexMkHfYKzf/M3lqyu6nAZ1XUNFOW1DtTFeemiX1XXsME+VXXtj2fcXh156qFMrriz+RUS6Zt9/jrp76xYuJKGugb++KO/UtArn7EH79zuMmoqaijs3TqkF/YpaGqrpboiPYyroHdBqz5FfQqa27YT1uutpLgowerKtoFhVUWS4qYAUlyUYPXq1jeAr1qVpLhwq/15jx4j0XQCI1Xb+nhN1daSyGkncOfmtBuoU7W1zQE9kZvT/vLaCfDaMh4frXV2OhsB/GoD7U8Dx3XyOgGIoqi0o7amsywlW2O9nSlKRaQaUyyfV91q+MlaQ8YU09iQYtHsCobskm5fNLuC5Jqouf+QMcVtbsxcMHN1u8sTTHl9FlNen9Vm+o8+fwQHjxnJWTfdz8KVGx7WM6tsKSV9e9ErP5fVNeliP3bYIGrr1/DRsvKmPssYO3wQj78xu3m+scMG8s+353Ti3nR//7rnBf51zwvNnweU9gMg6vj+5TbmvDWXI0/9JImMRPPwlJ33Hc3COYupqUz/oF304VJ22W80777yHgCJjAQ77DmSZ+76f520J92C9Xor2XNsDvX1EdPfqWeP3dKhcfo79TQ0wJ5j01fJxo3N4bW3Wv8i/p9p9ez5iextvr3dTSIvj8y+fWiYN5/cUSMBiJJJGsrKKNx3nzb9s0uGQmMjDQsXkj00PTylYeFCSCabh6tklwylYX7rGzMb5s8nu6Rb/1Pskjw+WuvsXy1WARsaZDkK6Hkj7TfTi3fOpXJ5+h9a1cp6Jl/7DhlZCUbs1afd/n2G5rHLJwfw+I2zqC5voLq8gcdvnMWYwwfSe0j6ktr+E4fx3r+XMfOZxSTXpJj5zGLee2EZB5zS8c2GWicjEZh0+nj23XHYRoVwgDf+W8aHS1fy/c9+ivycLAb3LuLc4w7i71Nn0NCY/o3+vhem8dkDxrL3DiVkZiQ45ZA9GDGwD4+8OnNr71K3tmzBCl546FW+eMnn6D2wF1nZmZx1zReoKq9m5ouz253nhb9PJZVM8aWrv0B2bjYjxw5j4oUnMPmWJ5v7/OO3TzLxwhMYOXYY2bnZfOnqL5Bck+SFh17dVrvWFazCer3RksmIuroUDU1DVOsb0p9Tqba/JY4YlsWxR+Xzg6uXs3xFkuUrkvzg6uV85tP5DC9N36D/9TN68fi/anhoShVr1kQ8NKWKJ56t4Zwze7VZntoqOvRgVj/7PA2LFpFqWMOqJ54iZGSQv8fubfpm9u1L3m5jWPnIoySrqklWVbPykUfJG7sbmX3SP28LDz6ImndmUT39baJkkurpb1P77iyKDjlwW+9at+Txsfk6+4z4M8B5IYQnoyj6Z8uGEMJhwLnAQ528zm7rg5eX89xtc2ioTZJbkEnJ7r34ym37UzwgfSls1aJafnXCv/nS7/Zl1D59AZh4/R5Mvu4dbjz2eQDGHJZ+jvhaw8f1YeKkPXjy/97jvoum0bskj1OuH0fpJ3pv8/3rjvYcNZQJ++5K/ZpGHr30rFZtB/7wN0D6KSm3nPNZPjvpDhavqiSK4ILbHuHSk4/kXz/+Og2NSR5/Yza/eOTfzfM+Pe19+hXl85PTx9O3MJ8Pl6zgvN8/zJJVHQ95UdoNZ93MN391Fn9891dEqYjZr83hkvHXNp/dPvLUQ/nO787hhOL0KIraqjouGX8tF9z8VR5c/kdqKmp59NanePCXjzYv8283Tia/KI+fPn0F+cV5vPefOVxy7HXUVW9X40Gt15vgzgcqOfs7627aLh79XwD++eBQdhiexe6HzeOxu4fyyQPTJ0XuuGkQF1y6jJ0O+giACf+Tfk7yWgfuk8tfbhrEpT9ZwWnnLmFkaSZ33DSI/fZ0KMTGKD7icFJ19Sy+5fdEdXVkDytl0DlfI5GTQ2N5OWWTbmDQOV8ld/QOAPQ/7VRWPvgQC66dBKx7jvhauSNHMOD0L1L+6BSW3XE3mX370P/0U8kZPjyO3et2PD42X4g25Zrvxy0shJHAq0A/4DVg7em+3YD9geXA/lEUfdRpK9247VpQUlJScvE/j6CyccW2XLXaUZTZj/N3uROAo6+8bYPjr7X1DepVyNNXfw2ALw47h+VlK2Peou1b/5K+/Dt6jLKysrINDeHYUl29Xs97YxSkFn/8DNp6EoNJDEyfUDjoD7eyuMpaHafBhYW8fPY5AKSWftLjoytIDGb43h9uUb3u1DPiURTNDSHsC0wifcf92ofyVgJ/BX7kH5CQpPhZryUpfp0WxEMIWcCuwMooik4LIQRg7XWGZVFnnnqXJG0267UkdQ2debNmAN4ATgaI0pY2vSzqktR1WK8lqQvotCAeRVEDsAToeU9bl6QexHotSV1DZz++8EHg802XOSVJXZf1WpJi1tmPL/w9cBfwZAjh/4A5QM36nbwBSJJiZ72WpJh1dhCfDkSkxx8etYF+GZ28XknSprFeS1LMOjuI/5h0YZckdW3Wa0mKWWc/R/yqzlyeJGnrsF5LUvw6+2ZNSZIkSRvBIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMWgU4J4CKFfCOG7IYSrQwgHtJj+lRDCv0MIM0IIt4cQhnXG+iRJm8d6LUldxxb/QZ8QwmDgNaCkadJlIYTTgN7ALcACIA/4CnBMCGGvKIqWb+l6JUmbxnotSV1LZ5wRv4h0ET8FOBB4E7gOOBc4PIqi4VEUDQBOBYYAP+iEdUqSNp31WpK6kM4I4scCf4ii6IEoil4FLgFGAZOjKPp/aztFUXQv8AAwoRPWKUnadNZrSepCOiOIDwdmtPg8s+n9tXb6vgKM7IR1SpI2nfVakrqQzgjiDUBWi8/1Te+V7fSt7qR1SpI2nfVakrqQziiyC4GWd9dXAl9m3ZmWlkYCyzphnZKkTWe9lqQuZIufmgK8ARy09kMURQ3AXzro++mm/pKkbc96LUldSGcE8R8Bgz6uUwhhIOmxiQ90wjolSZvOei1JXcgWB/EoiuYD8zei31LSl0AlSTGwXktS1+KNOJIkSVIMDOKSJElSDAzikiRJUgwM4pIkSVIMDOKSJElSDAzikiRJUgwM4pIkSVIMDOKSJElSDAzikiRJUgwM4pIkSVIMDOKSJElSDAzikiRJUgwM4pIkSVIMDOKSJElSDAzikiRJUgwM4pIkSVIMDOKSJElSDAzikiRJUgwy496AzhJCWLCB5sHbbEMkSRtkvZaktB4TxDfGfvn/pSG5OO7N2O5lZ6z7Odv/j/+BspUxbo36l/SFq78GwN2vvwMpj5FYJQYzfO+4NyJ+h/zjdBZXVcW9Gdu1wYWFvHx2+r9HXvUGhdbqWPUv6QtN34fHR9cwuLAQuGaLltFjgngURaUdtTWdfSnZhpsjSeqA9VqS0hwjLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMUgM+4NUNql5yzm30/V8Is7B7Pvofnt9llS1sgvrljOW1Nryc4OHPmZQs6/rB9Z2aG5z9/vWM29v19N+Yokw0dnccHl/djzgLxttRs9xqe/dDgnf+94Bu8wkLrqep6770Vu+fafOuw/6hPDOf+ms9lpnx2oWV3DY7c9w51X/61VnzOvOoXjvnoU+b3yef/1/3LTebczd+b8rb0r3dbVN67g2l+Wk5e77t/3Zz5dwD2/Hdxu//JVSS740TIee6aGEOC4o/K5edIAevfKaO7zwKNVXHH9Cj4qa2RkaSbXXNyPkyYUbvV9UfcXRRGrnniKqpenkqqrJbu0lH4TTyJ7yJB2+ydralj54EPUzHwXgPyxu9L38yeRkb+uHle/NY3yKU+QLC8ns09fek84loJxn9gm+9NTnHXN/3LAcXszYuwwZr/6Ad/91OUfO4/1euvwGNk8nhHvAp54sJK62miDfVKpiB+evZjiXgn+/vIIbptcwrRX67hl0ormPs8+VsVtN67kRzcOYMq0kUw4pYgffmUxSxY2bu1d6FFO/t5nOP2Kifz6vNv4bO8vceYO5/HUn5/rsH9eYS6TnriMmS/N5uQBZ3Px+Os49uyjOOk7E5r7TPz+CRzz5SO4ePx1nDzgbGa+NJtJT1xGbkHuNtij7uugfXKpmDO6+dVRCAc44/wlLF2e5P2XR/DeSyNYujzJWd9e2tw+9Y06zjx/Cddc3I9V7+3Aj3/YjzPOX8J/3qrbFruibq7i2eeomvoqg77xNYZd+2NyR41iye9uI1Vf327/5XfdQ7KyitLLLqH0sktIVlax/J57m9vr537Esrv+Sp8JxzL8+uvoPWE8y++6m/p5hr1NsWjOYv5y5X1Mue2Zjepvvd56PEY2j0E8ZksXNXL7L1Zy0aQBG+w3/dU6PprTwHmX9aOgKMHg0izO/l4fHruvkvr6FAAP31XBcROL2PPAPLKyAyed2YvSkVk88UDlttiVHiG/KI8zr/oCt3z7j8x4YRapZIq6mno+ePPDDuc59KQDSGQk+PPl99JQ18DcGfP4242TOfG88c19jv/mMTzw838wd8Y8Guoa+PPl95KZncmhn9t/W+xWj/fR/DU8/s8abriyP/37ZdC/XwY3XNmffzxZzbwFawC49Y7VjD8in89/ppCsrMDnP1PIMYfn87s7Vse89eoOKl94iV5HHE720CEksrPofdwxRI1JaqbPaNO3ceVKat+ZRd8TjyejsICMwgL6nng8tTNm0lhenl7eSy+Tv+sYCsbtQcjIoGDcHuSNGUPliy9v4z3r3p7883O88ujrrF5esVH9rddbj8fI5jGIxyiKIq6/aBlnnteHQSUbHiX0/rsNDB2WRe++6y6z7zouh7raiAX/TQeN999pYNdxrX9jH7NHDu+90/5vo2prt4N3Ia8wl9Kdh/Dn2b/m/sW3M+nxS9lhjxEdzjN6z5HMefNDUslU87TZr33A0NGDyS/KI784nyGjBjLr1Q+a21PJFHPe/JDRe43aqvvT3b05o55BY//LqH3nctq5i/lw3pp2+701s56cnMC4sTnN08aNzSE7G96a2QDAtJn17L9XTqv59tszh7febth6O6AeIVVbS+PKcrKHD2ueFjIyyC4toWFBWZv+DWULITOT7JKhzdOyS4ZCRka6ralPy+UBZA8fRkNZ2+Wp81ivtw6Pkc3XY4J4CGFBRy+g4+vZMXr4rgqI4IRTiz+2b01lisLi1l9XUa/05+qq9LCWmqq2fQp7Jaip2vCwF63Tq38RAAefuD8XHnEVp488lznT5vKTxy8lv7j9sfv5xflUrappNa2qvLqpLY+C4vR4t+pV1a36VJZXN7eprc9/ppAZzw9n8YxRvPCPUgLw6VPKqKpOtelbUZmiV1Hbcta7OIOKytS6Pi3GiwP07p1BRVXb5Wnr6m71OlWXPpmRyGt9vCby8kjVtx3alKqrJ5HbdhhDIi+PVF3duj7tLa/OoVJbk/V66/AY2Xw9Joh3N2UfreEvN63iouv7b1T//KIEVRWtA0Pl6vTngsL0zWz5hW37VK1OkV8YUPuOPPVQJlfc2fwKifT/q79O+jsrFq6koa6BP/7orxT0ymfswTu3u4yaihoKe7cO6YV9CpraaqmuqAWgoHdBqz5FfQqa29TW7mNyGDEsixACJUMy+cMvB1G2OMlLr7UtwsVFCVZXtg3UqyqSFDcF9OKiBKtXJ1u3r0pSXGgZ1IYlctNXUlK1rY/XVG0tiZx2wkRuTrthIVVb2xw+Erk57S+vnXCitPXr9YBhG/fzsyXr9dbhMbL5esxPoCiKSjt6AYvj3r71TX+tjopVSb52QhnH7z2X4/eeC8Dl5y7hhkuWtem/067ZLFqwhtXl64LErOn15OYFSnfISvfZLZtZ01sPQ5n1dj0779b6crzW+dc9L3BC8RnNr2nPzgQg2oSLCHPemsvovUaRyFh3OO2872gWzllMTWUtNRU1LPpwKbvsN7q5PZGRYIemS6TaOCGkX1E7X86eY3Oor4+Y3mIY1vR36mlogD3HZgPpoSqvvdX6+PjPtHr2/ET21t1wtdHd6nUiL4/Mvn1oaHGTWJRM0lBWRnZpSZv+2SVDobGRhoULm6c1LFwIyWTzpfjskqE0zG9901nD/Plkl7RdntLWr9fL5i/f5GVYr7cOj5HN12OCeHdzxIQC7n1+OH94rLT5BfD96wZwzg/7tum/x/65DN8hm99ct4KaqhRLyhr54y/LmXBKETk56a/xs6cXM+X+Sqa9WsuahoiH76pgwYdrGH9y0Tbdt+5s2YIVvPDQq3zxks/Re2AvsrIzOeuaL1BVXs3MF2e3O88Lf59KKpniS1d/gezcbEaOHcbEC09g8i1PNvf5x2+fZOKFJzBy7DCyc7P50tVfILkmyQsPvbqtdq3buX9yJctXpH/xXLKska9duJRB/TM4eL+2l4dHDMvi2KPy+cHVy1m+IsnyFUl+cPVyPvPpfIaXpn9R/foZvXj8XzU8NKWKNWsiHppSxRPP1nDOmb226X6peyo69GBWP/s8DYsWkWpYw6onniJkZJC/x+5t+mb27UvebmNY+cijJKuqSVZVs/KRR8kbuxuZffoAUHjwQdS8M4vq6W8TJZNUT3+b2ndnUXTIgdt617q1jMwMsnKyyMjMIATIyskiKyerw/7W663HY2Tz+BzxmOTmJcjNa/t7UK8+CYp7ZzDt1Vou+vJi7nhqGINKMkkkAtffPohfXLGczx3wEVnZgaOOL+Sbl/RrnveICYWsXJ7k2u8uo3xFkhE7ZvHTPwxm0FC/5k1xw1k3881fncUf3/0VUSpi9mtzuGT8tdRUpi+RHXnqoXznd+dwQvEZANRW1XHJ+Gu54Oav8uDyP1JTUcujtz7Fg798tHmZf7txMvlFefz06SvIL87jvf/M4ZJjr6OuumeNdetMdz9YyQWXLKO6JqJP7wSfPDCPp+4fSlFhgnkL1rD7YfN47O6hfPLAdDC/46ZBXHDpMnY66CMAJvxP+jniax24Ty5/uWkQl/5kBaedu4SRpZnccdMg9tuzZ13m1NZRfMThpOrqWXzL74nq6sgeVsqgc75GIieHxvJyyibdwKBzvkru6B0A6H/aqax88CEWXDsJWPeM5LVyR45gwOlfpPzRKSy7424y+/ah/+mnkjN8eBy712197/ff4NNnHd78eUrtPQAcnZgIWK+3JY+RzRPau8zb04QQFpSUlJQ8+PIIGpJd7qrndic7YzAHDHsRgC8OO4flZStj3qLtW/+Svvx1/q0ApJZ+ElIeI7FKDGb43h9SVlZW1jRUY7uytl4Pu/pyFldVxb0527XBhYW8fPY5gLW6K2hZqw/6w60eH13A4MJC5l95zRbVa4emSJIkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMTCIS5IkSTEwiEuSJEkxMIhLkiRJMQhRFMW9DZ0ihLBgA80liUSCfgMzgeS22iR1KIPsjAEArFy8ilQyFfP2bN8SGQn6Du6d/pBahsdI3DJYtKSRVCq1Joqi7Li3ZmvYmHqd2auYZKpn/HzqrjISgQH5BYC1uitoWauX1VR7fHQBGYlA4+qKLarX20sQHwo0Aku30eZsTYOb3hfHuhVay++ja+kp38dAYE0URQVxb8jWsJ3U657yb7Gn8PvoenrKd7JF9brHBPHtxdofYFEUlca9LfL76Gr8PtRV+G+xa/H76Hr8TtIcIy5JkiTFwCAuSZIkxcAgLkmSJMXAIC5JkiTFwCAuSZIkxcAgLkmSJMXAxxdKkiRJMfCMuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuCRJkhQDg7gkSZIUA4O4JEmSFAODuLQRQghzQwjPxb0dkqQNs16rOzGIq0sJIQwPIdwZQngnhLAqhFATQpgVQvhlCGFI3NsnSUqzXktbLjPuDZDWMwgYDkwG5gNrgN2BrwGnhBD2jKJoWYzbJ0lKs15LW8ggri4liqLXgMPWnx5C+DdwP3Am8PNtvV2SpNas19KWc2iKuouPmt57b0znEEKfEMLPQgjvhxDqQwjLQgj/DCEcvV6/w0IIz4QQKpouq74WQvjiRix/ZAghCiFc1U7bn0MI0XrTnmsatzgqhDC5aX0rQgi/DiFkhRBymy7nLgoh1IYQngghDFtvGVc1rXOXEMJPQwgLQwh1IYSpIYSD1+ubCCFcGEJ4O4RQFUJYHUJ4N4Rw88b8/5OkLWC9tl5rI3lGXF1SCCEHKAJygN2A65uapmzEvP2Al4CdgHuB/2tazoHA/wBPN/U7HngIWAjcAFQDpwP3hBBKoyi6oRN3CaAA+CfwDHARcBRwAdAA7ApkANcCI4DvAncAR7SznDuatvV6oBi4EHg0hDAqiqLVTX0uA64G7gR+Q/qX7tHA+E7eJ0nbOeu19VpbIIoiX7663As4C4havD4ETt3IeW9tmufMdtoSTe8ZpM/arAQGt2jPBd4A6oEhLabPBZ5r8Xlk0zquamcdf04fWq2mPdfU/9vrTX8NSAEPrDf9hqb+Y1pMu6pp2iNAaDH9803Tv9Fi2hvAlLi/R1++fPX8l/Xaeu1r818OTVFX9SRwNHASMIn0GYXeHzdTCCEBfAGYHkXRHeu3R1GUavrPfUjfZHR7FEWLW7TXkR7TmA1M2LJdaCMJ/G69aS8CAbilnemQPku0vpuiKGp5KfXZpvcdW0xbDYwNIXxiM7dVkjaW9TrNeq1N5tAUdUlRFC0CFjV9fCiE8CTwXAihIYqi2zcw6wCgFzDtY1Yxqul9Zjtta6ftsLHbu5EWRVFUv960VU3vH3UwvW87y/mw5YcoilaGEAD6tZj8I9JnYqaHED4kXfwfBR5p8cNNkraY9dp6rc3nGXF1C1EUPQ8sAL68sbNsxc35uOVndDA9uYF5OmoLm9M3iqKXSf9gOhl4AjgU+DvwUgghbwPbIUlbxHq9aX2t19s3g7i6kzygz8f0WUb67MS4j+n336b33dpp2229Pu1Z2fTe3vZ09pmZzRJFUVUURQ9GUXRuFEW7kD7rcgDpS8GStDVZrzeB9Xr7ZRBXlxJCGNTB9P8lfSlv6obmb7qMdy8wLoRwWjvLWXsW4g1gHvCVEMKAFu05wPdI3xn/2AbWUwksYb275EMIBwAHbWgbt4UQQv92Jr/Z9N6vnTZJ2iTW685hvd6+OUZcXc1PQwi7A0+RvvO9gPRjrE4mPQbxqo1YxqWkHzV1ZwjhWOBl0jfz7E+6mP8wiqJkCOF80o/Dei2EcBtQQ/pxWHsDFzWNe9yQ3wA/DiE8Sno830jgq8B0Pv4Mz9b2bgjhZeBV0o/7KgW+Sfomqofi3DBJPYb1unNYr7djBnF1NQ+Svnx4BukbeVKkC/wvgZ9FUbT04xbQdDPMgaSfzfo54BTSlz+nAbe36PePEML/AJcDF5M+HmYAp0VRdM9GbOv1pG/OOY30D5K3SD814CvEX9h/TvopAt8m/ezaJaRvALouiqINXcKVpI1lve4c1uvtWGj9VB1JkiRJ24JjxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQYGMQlSZKkGBjEJUmSpBgYxCVJkqQY/H8CRQgmubA4cwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 720x720 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = pt.tensor(\n",
    "    [[1., 3.-1j , 4.],\n",
    "     [3.+1j, -2., -6.+1j],\n",
    "     [4., -6.-1j, 5.]]\n",
    ")\n",
    "AH = A.conj().T\n",
    "\n",
    "assert pt.allclose(A, AH)\n",
    "vis.plot_matrices_as_heatmap([A.real, A.imag], [r\"$Re(A)$\", r\"$Im(A)$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Unitary matrix\n",
    "\n",
    "A unitary matrix is a complex square matrix whose column vectors form an orthonormal basis. The matrix product of a unitary matrix with its conjugate transpose, denoted by $\\dagger$, yields the identity matrix:\n",
    "\n",
    "$$ \\mathbf{Q}\\mathbf{Q}^\\dagger = \\mathbf{I} $$\n",
    "\n",
    "Example:\n",
    "\n",
    "${\\mathbf{Q}={\\frac {1}{2}}\\begin{bmatrix}1+i&1-i\\\\1-i&1+i\\end{bmatrix}},\\quad\n",
    "{\\mathbf{Q}\\mathbf{Q}^\\dagger={\\frac {1}{4}}\\begin{bmatrix}1+i&1-i\\\\1-i&1+i\\end{bmatrix}}{\\begin{bmatrix}1-i&1+i\\\\1+i&1-i\\end{bmatrix}}=\\begin{bmatrix}1&0\\\\0&1 \\end{bmatrix}= \\mathbf{I}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 480x480 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Q = pt.tensor(\n",
    "    [[(1+1j)/2.0, (1-1j)/2.0],\n",
    "     [(1-1j)/2.0, (1+1j)/2.0]]\n",
    ")\n",
    "QH = pt.conj(Q).T\n",
    "I = pt.mm(Q, QH)\n",
    "\n",
    "assert pt.allclose(I, pt.eye(2, dtype=pt.cfloat))\n",
    "assert pt.allclose(I, pt.mm(QH, Q))\n",
    "vis.plot_matrices_as_heatmap([I.real, I.imag], [r\"$Re(I)$\", r\"$Im(I)$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrix decompositions\n",
    "\n",
    "### Eigen-decomposition\n",
    "\n",
    "The eigen-decomposition of a square matrix $\\mathbf{A}$ is:\n",
    "\n",
    "$$ \\mathbf{A}=\\mathbf{Q}\\mathbf{\\Lambda}\\mathbf{Q}^{-1} $$\n",
    "\n",
    "$\\mathbf{Q}$ is a square matrix formed by the eigenvecctors of $\\mathbf{A}$, and $\\mathbf{\\Lambda}$ is a diagonal matrix whose diagonal elements are the corresponding eigenvalues."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1080x1080 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = pt.tensor(\n",
    "    [[0.0, 0.0,  2.0],\n",
    "     [1.0, 0.0, -5.0],\n",
    "     [0.0, 1.0,  4.0]]\n",
    ")\n",
    "\n",
    "# the eigen decomposition returns complex tensors; to compare them with\n",
    "# the original tensor, we remove the imaginary part\n",
    "val, vec = pt.linalg.eig(A)\n",
    "Q = vec.real\n",
    "L = pt.diag(val).real\n",
    "invQ = pt.linalg.inv(Q)\n",
    "\n",
    "assert pt.allclose(pt.mm(Q, invQ), pt.eye(3), atol=1.0E-3)\n",
    "assert pt.allclose(A, pt.mm(Q, L).mm(invQ), atol=1.0E-2)\n",
    "vis.plot_matrices_as_heatmap([Q, L, invQ], [r\"$Q$\", r\"$\\Lambda$\", r\"$Q^{-1}$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Singular value decomposition (SVD)\n",
    "\n",
    "The SVD is a factorization of a real or complex matrix that generalizes the eigen-decomposition of a square matrix to any ${ m\\times n}$ matrix:\n",
    "\n",
    "$$\\mathbf{A} = \\mathbf{U}\\mathbf{\\Sigma}\\mathbf{V}^\\dagger$$\n",
    "\n",
    "- $\\mathbf{U}$: is a $m \\times m$ unitary matrix; the columns of $\\mathbf{U}$ are called left singular vectors\n",
    "- $\\mathbf{\\Sigma}$ is a $m\\times n$ matrix containing a diagonal matrix of dimension $\\mathrm{min}(m, n)$; the diagonal elements are called singular values\n",
    "- $\\mathbf{V}^\\dagger$ is a $n\\times n$ unitary matrix; the columns of $\\mathbf{V}^\\dagger$ are called right singular vectors\n",
    "\n",
    "example:\n",
    "\n",
    "$\\mathbf{A}$=$\\begin{bmatrix}1&2 \\\\ 3&4 \\\\ 5&6 \\end{bmatrix}$=$\\begin{bmatrix}-0.23&0.88&0.41\\\\-0.52&0.24&-0.82\\\\-0.82&-0.4&0.41 \\end{bmatrix}\\begin{bmatrix}9.53&0\\\\0&0.51\\\\0&0 \\end{bmatrix}\\begin{bmatrix}-0.62&-0.78\\\\-0.78&0.62 \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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6cOguW3HdyavuJbn9uXdonlE9UGVnpFOwYu3DzpqlpnDPb48kNTmJSx7+F2XlNYeE5UUl/Ovjcbxw3RnMXbyM98dN+WU70cDlZCe+FC3Jq3qmevGSMjp1SK5Wf+KUYvqdNJM/39yWs0/OobQs4q4HFrPnET8y5p2uZGUmccJ5c0hPD8z4shsd2iXz0egVHHv2bNLSAocd+LOOx0YpJSsRsEqWVQ1VxflFNGtb/X2a+fZECmfmseP1B9VJ+9S4PPVgewbevJBd+s2grAzOPTWHrXqm0qZVzcFopx1W3T/Us3saf727HT12/YGJU0rYoscaJwfSAr/eJ5N/vbaMP961iLtvbLPm5pqsgrzEsPWs3KrHdHbLLBbMWlRD/UKyWlQPTFm5WRRWbKswbznNc6vWyW6ZRWFe9YCmOhOSkpJS01Obdyovq/n7h+KVlBxo1SZxBX3RnCWUl9V+v6nilZScRFG0nPLyn/+eN5qgFUVR59qWVZyd6VSHzaniP6O/5T+jv6183rFVNumpKWy+SRsmzkoMH9x8kzakpabw3Y81D6WCRBB74MJjWFqwgsse+TfFpdWHt6wpNTmJbu1b8v64X74fDVmLnGS6dUnhszFF/GrnxNnS0tKIr8YX1zgEaOw3xWQ0S+KCMxMzi6UTuOKCltz658WM+7aYvjs247OvVvDk/e3ZpEPiMNqzbwZ79s1gxJsFBq3VpDZPJ6NDNku+nUur7RIXPspLy8mbtIDO/basVn/eqOksm76YN44ZCkBUmujg3jhmKNteuAddDtmq7hqvShtzH7u6dm1SePL+VcNU5y0o5e6HlrD/Xut3FSSp4iJWtJbvkCUlMH2m92itrjCvkNlT57HlLj2Y8Mn3QOJLyma9u/HW0+9Xqz95zDTSmqXRvVdXpn49HYDuvbqSlp7KpDHTKutsuXPPKuttsVOPyuXaMNZ1rHfs2LHTvlsMYMHcvLpsVpPVpn0OT78zCICTu5zPgpnVT1xow2jTsRUfRCOYOXPmz76C69iHejB7UT4fjJvC5b/Zm9ysZuRmNePy3+zNe2MnM2dxfo3rtM7J5PHLTmDO4nwu/+vLNYasY/fsVTmZRlazNC447Fd0bJXDqO+mb9D9aSguOLMF9zy0hHHfFrF8eTk3Dl5Eaiocc2j16dh33iGd4pKIvz61lNLSiBUryvm/vy6heVZgi80S07vv2TeDJ57JY96CUqIo4pPPV/D+J8vZcft1D/9sarodsx2Tnx1D3pSFlBWV8t3QT0lKSaLjXtXv3dj24j3Y7+lT2efxE9nn8RPZYeB+AOz96PF03Dcxy2ZUHlFWVEp5SeI4KC8tp6yolMgzfU3ed5OKmbegFIBJU4s57aK57LdHBgfuXT1orVhRzvOvLmNpxZXuaTNKOP8P89lp+3Q2rzjOXxyxjK/GF1FSElFcHPHiiGX8/cV8Dj3AkylreuWh/3L8FUfSbdsupDVL44ybTqSspKzG6djnTV/AqBFfcP7g/uS0ziandTbnD+7Pxy9/xvwZiROQI/76Jrse2oc9jt6V5JRk9jh6V3Y5pA+vPvxGXe+aJP0sjeaKVkNz7d9e5+oT9ueVm84B4INxU7j9uXcql68cbrj75Q8AcNye27NFp7Z0aZPL+3dfWFnvi0kzGfCXlwDYpkt7fnvwbuRkNmN5cQkTZy1gwF9eWutVsqbkigtyyV9WzkEnzCIvv5ydd0jnP8M3oXlWEtN/LGG7faYz4u+bsNduGWzaJZWXhnbkprsXcvVtCwHotVUaLw/bhJa5iaGGT/xfOwbevJA+B8xgWUE5HdqlcNlvcznnFGd5XFOPk/pQWljCJ5f/m5KCEnK3bEvfwUeQkplG4dx83us/nL53HUHrHTYhLbsZrHaRsWBm4gpkszbNK39Ha+FXs/j40n9V1nmv/zMA9L56f7ocsnWd7Zc2Ph+OXsGNgxeyaHE5rVslc9LRzbnpD6tmp8zpMZmH7mrHqcdmU14OQx5dwu/+MI+ioog2rZLpt18mQ+9rR1LFpa25C8q49vaFzJxdSkpKoHvXFO6+oU3l1W6t8s+7XyYzO4M73/wjmTkZfP/ZZK4+5DZWFKygbZc2PD7+z1xz6G2MG5kY4XFn/yEMGHIOwyYl/p/75NXPGTLgscrtTRg1kTv7D+HsP53CNcMvZc60+dxx+v18/9nketk/SfqpQrS28RGNRAjhx06dOnXa5ow/Mm9J9am8tWG0y23OG7f9FoDyeXtB+Zx6blETkdSBpHYfAHDWqOtYWLykftvThLROy+XNY59k5syZM9c2/KaxWdnHTv+iu8d5XVrtWHdIUd1p06lyOFGTOs5h1bHu0MG649DB+hPHse7QQUmSJEmKmUFLkiRJkmJm0JIkSZKkmBm0JEmSJClmBi1JkiRJiplBS5IkSZJiZtCSJEmSpJgZtCRJkiQpZgYtSZIkSYqZQUuSJEmSYmbQkiRJkqSYGbQkSZIkKWYGLUmSJEmKmUFLkiRJkmJm0JIkSZKkmBm0JEmSJClmBi1JkiRJiplBS5IkSZJiZtCSJEmSpJgZtCRJkiQpZgYtSZIkSYqZQUuSJEmSYmbQkiRJkqSYGbQkSZIkKWYGLUmSJEmKmUFLkiRJkmJm0JIkSZKkmBm0JEmSJClmBi1JkiRJiplBS5IkSZJiZtCSJEmSpJgZtCRJkiQpZgYtSZIkSYqZQUuSJEmSYmbQkiRJkqSYGbQkSZIkKWYGLUmSJEmKWUp9N0CSJEn6qZrnNOPCa4+k7z5bEUURn77/HQ/e+jIF+StqXSc1NZlTLzyA/Q7fgRa5WSxdUsCwIW/x9stfAnDWZf3Yde8tabdJLkXLixk7eiqP3fM6C+YsravdahCa52YxYMg57Hb4TkRRxKgRXzBkwGMULC2ssf7+p+zJ7x8+v0pZWrNUpo2fwe/6/AGA3HYtuODeM+lzwHakpqcyfcJMHr/674x9/5sNvj8bikFLkiRJDc7AO04gNS2Fsw6+G4Cr7z6JK/90HDdd/HSt61z751NIS09h0DmPM3v6Ilq0yiI7J2NVhSjinmufZ9rEuaQ3S2XA9Udx04Onc9GxD2zo3WlQBj11CanpKfTvOQCAa5/5PQOfHMANx9xVY/13ho/kneEjK58npyQzfPrDvPX0+5Vllzx4LrntWnBeryvIX7SM3/z+UG55ZRCndbuQ/MXLNuwObSAOHZQkSVKD0q5jLrvusxV/Hfwf8pYUkrekkL8O/g+/2n8b2nZsUeM6vfv2oM+venLnVf9g9vRFACxdVMCP0xZU1hn6f28w6ZtZlJaUUZC/gn8+8T49ttqE5jnN6mS/GoJ2XdvQ97AdeeTKYeQtzCdvYT6PXDmM3Y/ahbZd2qzXNvY6ti+ZORn894l3K8s26dmBD174hKUL8igvL+fVR94iMzuDTpt32FC7ssEZtCRJktSgbLZVR4qLSpj63ZzKsqnfzaG4uJQeW3WscZ0+u/dkzszFnHDO3gx/bxBPvTWQy289lpzczFpfZ8fdezJ35mKW5dU+HLGp6dG7G8Uripky9ofKsiljf6C4qISevbut1zaOuKAf/3vuoypXqp6769/sftQutOqQS3JKMkdd1I+Zk+YwZez0uHehzhi0JEmS1KBkNU+nIL+oWnlB3goys2q++tQiN5NNe7RLDDc85B4uPvEvtGmfwx/uOL7G+n1268FpFxzA/Tf/K86mN3hZOZk13otVsKSAzNWHYdai27Zd2H7vbXj5of9WKR//4bcUryjhuVmPMqLw7xx7+REMPvMBilcUx9b2utZo7tEKIfy4lsUdAFp9s5zy+TXfpKf4tWqbXPnv/T4/hrnL8+uxNU1H+4xs/ndI4t8rjpvP8pmL6rdBTciKTiX13YQNZn362FN32oYFMxvuEI+Gpk2nVjwzI/Hv5Ja5JHvCvU4kt8yFRtytrs+xXh/2O2wHLrnx6MrnD976MlnZ6dXqZeU0o7Cg5oOhsKCI8vJyHr/ndYpWlFC0vISnHniLe54+n/RmqRStWNWH77rPlgy84wTuGvQPPh85Mfb9aUjWnMhiyIDHyGpR/SpgVm4WhXnL17m9Iy7sx3ejJ/H9Z5Mry0IIDH77Br5+fwK/aX0WBXmF9D1sR24bcQ1X7HsDk7+aFsu+1LVGE7QkSZLUOL074iveHfFV5fN2HXNJS0+l+xYdmPp9Yvhg9y06kJaWwuRvZ9e4jUnfzKqxPIqAsOr5foftwIDrj+JPVzzD5x827ZAF1SeyaNe1DWnN0ujeqytTv04M6+veqytp6alMGjNtrdvKaN6MA07di4d+P7RKefOWWWzSowM3H39P5XDCj1/+jFmT57Bzvx0abNBqNEMHoyjqXNsfMGedG5Ak1co+VmoaGsqxPm/2Ej7937ece+Uh5ORmkpObyblXHsIn705g/uyap2L/8O3xLJybx5m/P4jUtBSyW2Rw2kUHMPqD7yhanriadcQpu3HhtUfyxwuHGbJqMW/6AkaN+ILzB/cnp3U2Oa2zOX9wfz5++TPmz1iw1nUPPH1vykrKePfZj6qU5y9axg/f/MhRFx1MZnYGIQT6HrYjm27bhe8/n7Ihd2eDajRBS5IkSU3HXVf/k7wlhQx9/UqGvn4lSxcXMPjqf1Yu3++wHXhp9A2Vz4uWl3D1eUPZtEc7/vHhtTz8r0uZP3spd1/zfGWdi649kozMNG595AxeGn1D5d+2O3ary13b6N3ZfwhLF+QzbNIDDJv0AEvm53HnGUMql+9/yp68nPdUtfWO+F0/3vjbezXed3XD0XeS0zqbJycO4aXFT3LuHafx4CVP8OXbX2/QfdmQHDooSZKkBmfZ0uXcOfC5WpevOdwQ4Mep87nmvKG1rAEHb3tNbO1rzPIXL+P20+6rdfmaww1X+u0OV9S6zsxJc7jxN4Njad/GwitakiRJkhQzg5YkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSJMXMoCVJkiRJMTNoSZIkSVLMDFqSJEmSFDODliRJkiTFzKAlSZIkSTEzaEmSJElSzAxakiRJkhQzg5YkSZIkxcygJUmSJEkxS4lzYyGEDCA7iqJ5q5W1AS4EWgPPRFH0SZyvKUlNhX2s1Ph5nEuNR6xBC3gI6APsABBCSAM+BDavWP67EMLeURSNivl1JakpsI+VGj+Pc6mRiHvo4B7AK6s9P55Ex3AysA3wAzAo5teUpKbCPlZq/DzOpUYi7qDVAZi22vNDgS+jKHouiqJvgceAXWJ+TUlqKuxjpcbP41xqJOIOWmVUHY64L/Duas8XkBhfLEn66exjpcbP41xqJOIOWhOBgwFCCHuQOCvz9mrLuwCLY35NSWoq7GOlxs/jXGok4p4M4y/A4yGEcUAnYCrw1mrL9wK+jvk1JampsI+VGj+Pc6mRiDVoRVE0NIRQBhwFfAr8KYqiEoAQQmugJYkORJL0E9nHSo2fx7nUeMR9RYsoioYBw2ooXwjsFPfrSVJTYh8rNX4e51LjEOs9WiGEs0IIm8a5TUlSgn2s1Ph5nEuNR9xXtB4HohDCdBIz5LwHvBtF0YyYX0eSmiL7WKnx8ziXGom4g9auwP4kpiI9DjiTRGcxlURn8S7wXhRFs2J+XUlqCuxjpcbP41xqJOKeDOMz4DPgrhBCMonOYl8SHcbJwNlAFPfrSlJTYB8rNX4e51LjEffvaFWKoqgMmAxMITE16XwgkPghPknSL2AfKzV+HudSwxbr2ZCKaUf3Bfar+NsKKAc+B54F3gFGxvmaktRU2MdKjZ/HudR4xH3ZeS6Jy9lfAa8BA4H3oyjKj/l1JKkpso+VGj+Pc6mRiDtorRyKmA6kVfwlx/waktRU2cdKjZ/HudRIxB201rzcPQAoDyGMJTFLzjt4VkaSfi77WKnx8ziXGom4Zx1cDLxU8UcIoS2rZso5Cvg9UEriLI0k6Sewj5UaP49zqfHYYLMOhhBSgM2BLSr+OpGYKSd1Q72mJDUV9rFS4+dxLjVscc86uCurLnXvAWSS6BAmAcNY9QvnkqSfyD5Wavw8zqXGI+57tD6peJwKPMeqXy+fGfPrSFJTZB8rNX4e51IjEXfQOgt4N4qi6TFvV5JkHys1BR7nUiMR92QYf4tze41Z8+xmXHzZwey2e0+iCEZ9PIn7732NgmVFta7z1sjrKCoqobwsqiy75HdDmTplPgDn/m4/+v5qc9p1aMGK5cV8NWY6j/7lbebPy9vg+9MQRFHEwufeY8lbn1NeWESzzTrS/reHkd61fbW6pUuXMX/Ymyz/5gdK8wpIzs4kZ89etD5xX5JSqx82+Z9+y6w7nyV7715scumxdbE7DU7z3CwGDDmH3Q7fiSiKGDXiC4YMeIyCpYW1rrP/KXty0lXH0G7TNizPX8H7z3/MY1c9TUlxKalpKVx431n03m87WnVsScHSQkaN+JzHrx7OsiUFdbhndcc+dv31v/EEDj33ADJbZDLx8ykMuegxpo2fUWPd9fls7nXsbpx5y0m037Qtc6fNY+h1zzDypU/rancajNMGHsEhp+9JVnYGE8dO54GBw/nh21k11r3rX5ez9S49KC0prSx7/KYXeXXo/wDovm1nzr7+GHr06kKrdi24+tg/8+X739bJftQnj3Op8dggk2GEEI4JIQwLIYys+BsWQjh6Q7xWQ3X19UfRslUWp5/4IP1PepCWrbK46toj17ne9Vf9gyMOuqvyb2XIAogiuOtPL3PsYfdw9mkPQxRxy50nbMjdaFAW//sjlr7zJV2uP52eQweSsVVXfrzlacqXVw+35cuLSdukNZ1v6M/mT11Nlxv6s+zz71nw1JvV6pbmFTB/6OtkbNWlLnajwRr01CW0bN+C/j0HcMbmF9OyfQsGPjmg1vqbbb8pVw27mL/f9gLHtDyTS/e4lp0P2oHTbzgegOSUZPIXF3DDMYM5puWZDNh1EJ16duTKJy6sq12qN/axa3f8lUfS76z9GHTwbRzX9hzGf/Qdt79+Hc2ymtVYf12fza127cmgpy5m6HXPcFSL/gy9/lkGPX0pW+y0WV3tUoNw3EUH0e+U3bn2hPs4Yasr+ObTSdz2j0tollX75HjPP/AGx3S7tPJvZcgCKC0u5cNXv+SGUx6si+ZvdDzOpYYv1qAVQkgNIbwMPA+cBmxb8Xca8EII4d8VM+g0ae3at6Dv7pvz8ANvkrd0OXlLl/PwA2+y+15b0q59zs/e7uOPvMvE7+ZQWlpOwbIinvv7x/TcvAPNs2v+ctHULPnvaFoduTvpm7YnKT2VNifvR1RaRv6o6mdI0zq0ovWxe5PWoRUhKYm0jq1pcUAfCsdNq1Z37sOvkntYX1I7tKqDvWiY2nVtQ9/DduSRK4eRtzCfvIX5PHLlMHY/ahfadmlT4zodN2tPwdJC/vePj4iiiHnTFzDqP1/Qo3d3AFYUFvHENcOZPuFHysvLWTRnCf964DV677ddXe5anbKPXT9HXNCP5+95hWnjplO8opgnr3+WlLQU9jxm12p11+ezefj5B/Hpa2MY+eIoykrLGPniKEa/PobDL+hX17u2UTv8rH14/i9vMm3CLIpXlPC3218mJTWFPQ7t/bO2N2PiHF5/eiQTv/oh3oZu5DzOpcYj7itaVwOHA48AnaMoahlFUUsS05E+BBwBDIr5NRucnpu3p7iolCmT5lWWTZk0j+LiUnr0rD6MbXWDrj+KF0dczkOPn8OhR/RZa92ddt2MObOXsCx/RSztbsjKClZQMm8JzTbvVFkWkpNJ796Boqmz12sbhWMmk969Q5Wypf/7irK8Aloe2jfW9jY2PXp3o3hFMVPGrvrCNGXsDxQXldCzd7ca1/nsv2OYOXEO+5+yJ0lJSXTcrD27Hb4zI18aVevr7HTQDkz6cmrczd+Y2MeuQ2ZOJh27t+PbTydVlpWXlTP5y6n06NO9Wv31+Wz26N2N70ZPrLLe959NqvWz2xRlZjejw6Zt+P6LaZVl5WXlTP56Bj16da11vcPO3Jt/TryXRz+6ibOuP2atV7+aEI9zqZGI+4zIqcBLURRVGbsTRdFsYEAIYRPgdODWmF+XEMKPa1ncYS3L6lxmVjoFBdXDT8GyFWSu5T+ZP1z6NOO/Tpy933Hn7lx9w9EkJwde+dcX1eruuHN3Tj9rL26+7vlY295QrRwemLTG0KHkrIwahw6uacE//8eKqXPY9K7zKstKFuax4Om36HLTmYSkDfaTdI1CVk5mjfdiFSwpIDMno8Z1ipYX89rjbzNgyDkMfHIAySnJvPG39/jvE+/WWP/A0/fmwNP25rK9ro+17RsZ+9h1yKr4PBWscZ9e/uKCymVV66/7s5mZk8GyJVXr5C8uIDMnM65mN3iZ2Yn3atka7+WypYVk1jKqYuht/2LGxDkULF1Ot206ccWQM+jQtQ23n/foBm/vRq7ejnNYv2O9bOJUymYu2hAvrzWU5a0aLfP3z7+B8jn12JomJqkDXXf8hZuIpyWVNgXeWsvyNyvqNCn7/3o7XnljYOVfCIGsGu4VyGrejMKC2r/0f/n5NIqLSyktLefTTybz0j9Hc2C/XtXq9d29J3+85VjuuOXfjB41JdZ9aaiSMhIBtnyNgFtWsLxyWW0WPPMOS9/8nK43n0lq6xaV5XP+8m9aHvEr0jZpHX+DG7j9T9mTl/OeqvwLSYGsFtW/lGblZlGYt7zGbfy6/z6ce8ep3HD0XRySfjInbnIeOa2zGfT0JdXqHnregVxw75lcc8htVa5MNEL2setQUPF5ysrNqlKe3TKrclnV+oXr/GwW5i2neW7VOtktsyjMq30il6amMD/xXjVf471s3iKTwlpGVUwYPYVlSwqJooip43/kr9f9kz0P70Nasyb/W7we51IjEfcVrSVA9bEZq3QHlsb8mgBEUdS5tmUVZ2c61bZ8Q3vnzXG88+a4yuft2rcgLT2F7j3aMXVyYvhg9x7tSEtLYfKkueu93fLyiBBClbL9f70dl15xMLf88UU++9SQtVJyVjNS2+WyYtJMMrZMTFoRlZVRNHUOOXvvUOM6URQx77H/UDBmEl1vPYvUdi2rLC8cM5kVk2ax6MUPAChfUQzApC8nsdmjV9Q4O2FT8c7wkbwzfGTl83Zd25DWLI3uvboy9evEjMXde3UlLT2VSWOm1biNLXbuwdfvT+DrDyYAsGjOEv7z6Ftc++xlVeqdOPAojr38CAb++mYm17KtRmQJ9rFrVZhXyOyp89hylx5M+OR7AJKSk9isdzfeevr9avUnj5m2zs/m5DHT2HLnnlXW22KnHrV+dpuiwvwVzPlhAVv02ZQJnyX+70lKTqLHdp1555+frGPthPIoMaPuGv+tNUVLqKfjHBrOsS41BHFf0XoLuCiEcMCaC0II+wAXAm/E/JoNzry5Sxn10UTOv+hAclpkkNMig/MvOpCPRn7PvLk1T8Xec4sObL5lB1JSkkhKDuy0y2b85oRdeeet8ZV1jvrNzlx8eT+uHficIasGuf12YdHLH1E0fS7lRSUsePY9Qkoy2X23qlY3Kitj9n0vUjhuGl1vPbtayALY7K+X0e3e37HpPYm/5jtvSdaOm7PpPb9r0iGrJvOmL2DUiC84f3B/clpnk9M6m/MH9+fjlz9j/owFNa7z9QcT6LX31mzzqy0AaNEmh0POOYCJn6/6bJ97x6kcffGhXLnvDU0hZIF97Hp55aH/cvwVR9Jt2y6kNUvjjJtOpKykrMbp2Nfnsznir2+y66F92OPoXUlOSWaPo3dll0P68OrDTf6truLVof/j2AsPYtOtNiGtWSqnX3UEpaVlfPifMdXq5rbNZqf9tyU9Mw2ATbfsyG9vPp5PXh9L0fKSynqp6Smkpif60+TUZFLTU0hKbvRDtT3OpUYi7m+D1wEHAW+EEEYDK1PANsCuwIKKOk3eHbf8m4svP5innrsIgE8+SvyO1kr7/3o7LvvDoRxx0F0AtGmbzXkXHEC7djmUlZUzd+5SnnjkXV7996r7sy6+/GBKS8u4/e6Tq7zW1Vc+w7ixNf9+TFPS8qjdKV9exIybhiV+R6vHJnS+7jSSMtIpmb+Eqb9/kM7XnkbmNpuy/NsZ5H/wNSE1mSkD7q+ynS3+fi1AlWGEACE9FcrLq5Ur4c7+Qxgw5ByGTXoAgE9e/ZwhAx6rXL7/KXvy+4fP58ic0wF4/58f07pjS654/EJab9KS4uXFfP3+BG4/7T4gcZXsxIFHU1JcygOf3lHltc7Z9rJaA1wDZx+7Hv5598tkZmdw55t/JDMng+8/m8zVh9zGioIVtO3ShsfH/5lrDr2NcSMTM46u67M5YdRE7uw/hLP/dArXDL+UOdPmc8fp9/P9Z5PrZf82Vs8/+AYZzdO5/YXfk9k8g4lf/cB1J97PioIi2nZqyV8/vJHrThrC+E8mkZaeSv+rjqBzzw4kJQcWz8vjw1e/ZPg9Iyq3175La/72xZ8qn9/6bGLY8NN3vcLTg1+t8/2rQx7nUiMRoihad62fssEQugK3k5gVp3lFcT7wCnBNffzSeQjhx06dOnXac5c/sGB+fl2/fJPVpm02z750KQD7vHYvc5f73teF9hnZ/O+QywE4ucv5LPCG5TrTplMrPohGMHPmzJlrG37zS2zMfexe4TA/b3WoTadWPDPjEQBO2/4qFsxeUr8NaiLadMzlvUV/b3LHeUW7PNbr2OrHefm8vZwMoy4ldaDrjlN/0bEe2xWtEEIqsDWwKIqiU0Pi5qG2FYvnR3EnOklqQuxjpcbP41xqXOIc6ByAL4DjAKKEeRV/dgyS9MvYx0qNn8e51IjEFrSiKCoG5gLlcW1TkpRgHys1fh7nUuMS99Q9LwDHhjXnHJckxcE+Vmr8PM6lRiLuWQf/CjwN/DeEcB8wGaj2i471dROnJDVw9rFS4+dxLjUScQetsUBEYoxxtd9/WE1yzK8rSU2BfazU+HmcS41E3EHrZhKdgyQpfvaxUuPncS41ErEGrSiKboxze5KkVexjpcbP41xqPOKeDEOSJEmSmjyDliRJkiTFzKAlSZIkSTEzaEmSJElSzAxakiRJkhQzg5YkSZIkxcygJUmSJEkxiyVohRBahxAuCyHcFELou1r52SGED0II40IIj4UQusTxepLUlNjHSk1DCGGbEMIJIYT9Qgg1/tZpCKFXCOGPdd02ST/dL/7B4hBCB2A00Kmi6LoQwqlALvAX4EcgAzgb6BdC6BNF0YJf+rqS1BTYx0pNQwjhURLH8UrTQgjnRFH03hpVtwduAG6uq7ZJ+nniuKI1kMR/+CcAuwFfArcBFwL7RlHUNYqitsApQEfgDzG8piQ1FfaxUiNXcfLkHOB94FLgbqAl8GYI4az6bJukny+OoHUI8HgURc9HUfQpcDXQHXg5iqL3V1aKouhZ4HngsBheU5KaCvtYqfG7CPgwiqL9oih6IIqiq4BtgVHAoyGEC+u3eZJ+jjiCVldg3GrPx1c8jq6h7idAtxheU5KaCvtYqfHbAvjH6gVRFM0G9gdGAENCCL+vh3ZJ+gV+8T1aQDGQutrzoorH/BrqFuBMh5L0U9jHSo1fMrBizcIoiopDCL8BngHuCSGkATPrunGSfp44gtYsYPWZrvKBs1h11nV13YD5MbymJDUV9rFS4zcV6FPTgiiKykIIJwFPA7dT89VsSRuhOM58fgH8auWTKIqKoyj6WxRFc2uoe1BFfUnS+rGPlRq/d4FjQwipNS2MoqgcOJVE2Nq1Lhsm6eeL44rWNUD7dVUKIbQjcZ/B8zG8piQ1FfaxUuP3NLAJsBOJey2riaIoCiGcCSwAdqy7pkn6uX5x0IqiaAYwYz3qzSMx3EWStJ7sY6XGL4qiL4GT16NeBFyx4VskKQ7eNC1JkiRJMTNoSZIkSVLMDFqSJEmSFDODliRJkiTFzKAlSZIkSTEzaEmSJElSzAxakiRJkhQzg5YkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSJMUspb4bIEmSJP1UzXOzGDDkHHY7fCeiKGLUiC8YMuAxCpYW1lh//1P25PcPn1+lLK1ZKtPGz+B3ff4AQG67Flxw75n0OWA7UtNTmT5hJo9f/XfGvv/NBt+fhiCKIm66exGP/T2PpXnl7LR9Og/c0ZbttkqvdZ0nn8vjzw8vYcoPJWRlJnHi0c2579a2AAz7Rx6PPp3HhInFhAC9tkrjlkGt2WPXjLrapQ3KK1qSJElqcAY9dQkt27egf88BnLH5xbRs34KBTw6otf47w0dyZM7plX/HtDqTpQvyeevp9yvrXPLgubTt0przel3BsW3O5oMXPuaWVwaR3bJ5XezSRu+eh5Yw9Nl8XntmE+aN787uuzTjkJNnsaygvMb69z68mFvuWcQDt7dl8febMfnTTTnjhOzK5csKyrnuspZMHd2NWV9156iDm3PoKbP4cVZpXe3SBmXQkiRJUoPSrmsb+h62I49cOYy8hfnkLcznkSuHsftRu9C2S5v12sZex/YlMyeD/z7xbmXZJj078MELn7B0QR7l5eW8+shbZGZn0GnzDhtqVxqUh55cyhW/y6XX1ulkZCRx81WtKS6OeOk/y6rVzcsv58bBi/i/W9uy124ZpKQEsjKT2HH7ZpV1Ljwrl377ZZHdPInU1MClv80lOTkwesyKutytDcagJUmSpAalR+9uFK8oZsrYHyrLpoz9geKiEnr27rZe2zjign7877mPyF+8KiQ8d9e/2f2oXWjVIZfklGSOuqgfMyfNYcrY6XHvQoOzNK+MaTNK2aXPqmGCKSmB3tulM2ZcUbX6H41eTkFhxMQpxWy5+w902G4qB580k6/GV6+70qgvVrCsoJwdtq19KGJD0mju0Qoh/LiWxZ6GkKRfwD5WahoayrGelZNZ471YBUsKyMxZ9/093bbtwvZ7b8MjVw6rUj7+w2858LS9eW7Wo5SVlpG3aBk3/WYwxSuKY2t7Q5WXnxgemJuTXKW8ZW4yecuiavUXLErU//drBbzzQida5SZx0z2LOPSUWXzzQVdarLGdGTNLOPn8OVw1oCWbbZq6gfaibjWaoLU+wmffEGYuqu9mNBmhU6vKf7d6IYOyhWX12Jqmo1XrDDgk8e9mz7clo7hxdFYNQbO0XDi2vluhpqhs8RLKFvr/W10oa7buOorfmhNZDBnwGFktMqvVy8rNojBv+Tq3d8SF/fhu9CS+/2xyZVkIgcFv38DX70/gN63PoiCvkL6H7chtI67hin1vYPJX02LZl4YqJzsxEG5JXtXvc4uXlNGpQ3Kt9Qdd0pJOHROR47arW/PA40v5aPQKDjkgq7LupKnF9DtxFscf2ZxbBrXeULtQ5xpN0IqiqHNtyyrOznSqw+ZIUqNiHys1DRvrsf7O8JG8M3xk5fN2XduQ1iyN7r26MvXrxLC+7r26kpaeyqQx09a6rYzmzTjg1L146PdDq5Q3b5nFJj06cPPx91QOJ/z45c+YNXkOO/fbockHrRY5yXTrksJnY4r41c6Jq4alpRFfjS/mtOOyq9Xvs10aACGsfbtjvynikJNnccEZLbju8lZrr9zAeI+WJEmSGpR50xcwasQXnD+4Pzmts8lpnc35g/vz8cufMX/GgrWue+Dpe1NWUsa7z35UpTx/0TJ++OZHjrroYDKzMwgh0PewHdl02y58//mUDbk7DcYFZ7bgnoeWMO7bIpYvT0x2kZoKxxxafVbGLp1SOergLO64fzFz55dSVBTxxzsX0TI3qXL69o9GL2f/Y2cy8KKWjS5kgUFLkiRJDdCd/YewdEE+wyY9wLBJD7Bkfh53njGkcvn+p+zJy3lPVVvviN/1442/vVfjfVc3HH0nOa2zeXLiEF5a/CTn3nEaD17yBF++/fUG3ZeG4ooLcjnjxGwOOmEWbbeZyshRy/nP8E1onpXE9B9LyOkxmQ8+WTV088n729Ojeyrb7DWdzn2m8sXXRbz2zCaVwwqvv2MRS5aWc/2dC8npMbny7/b7GsdQ6EYzdFCSJElNR/7iZdx+2n21Ll9zuOFKv93hilrXmTlpDjf+ZnAs7WuMQgjcNLA1Nw2sfh9V186p5E3uUaUsJzuJx//cnsf/XPP23n6hcY8694qWJEmSJMXMoCVJkiRJMTNoSZIkSVLMDFqSJEmSFDODliRJkiTFzKAlSZIkSTEzaEmSJElSzAxakiRJkhQzg5YkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSJMXMoCVJkiRJMTNoSZIkSVLMDFqSJEmSFDODliRJkiTFzKAlSZIkSTEzaEmSJElSzAxakiRJkhQzg5YkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSJMXMoCVJkiRJMTNoSZIkSVLMDFqSJEmSFDODliRJkiTFzKAlSZIkSTEzaEmSJElSzAxakiRJkhQzg5YkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSJMUspb4b0FQ1z81iwJBz2O3wnYiiiFEjvmDIgMcoWFpY6zr7n7InJ111DO02bcPy/BW8//zHPHbV05QUl5KalsKF951F7/22o1XHlhQsLWTUiM95/OrhLFtSUId7tvHLzkrnsvMOZPedexAR8fFnU7j3r2+xrLBonetuuVl7HrnzVL6ZOJsLr3mmsjw1JZmLz96P/ffYkrTUFL4cP4N7Hn6TeQvzN+SuNBhRFPH90E/54ZVvKC0opsUWbel1+T7kbNZ6reuVFBTzv7OeZfmcfA575wKSUhLnhuZ/PoNJT33O0kkLKMkrYv/hp5LVObcO9kQNRf8bT+DQcw8gs0UmEz+fwpCLHmPa+Bk11l2f/nivY3fjzFtOov2mbZk7bR5Dr3uGkS99Wle702D4vkvSKl7RqieDnrqElu1b0L/nAM7Y/GJatm/BwCcH1Fp/s+035aphF/P3217gmJZncuke17LzQTtw+g3HA5Cckkz+4gJuOGYwx7Q8kwG7DqJTz45c+cSFdbVLDcb1vz+MVrlZnHjBo5x0wWO0ys3i2ksPWed6aanJXHPJIYwZ/2O1ZRefvR/bb92Zc658iqPPeYj8/BXccc0xhLAh9qDhmfzsl0wf8S273X0k/V45h1a9OjLqylcoLSxe63rjh3xA8y651cpTmqXSud9W9Ln2wA3UYjVkx195JP3O2o9BB9/GcW3PYfxH33H769fRLKtZjfXX1R9vtWtPBj11MUOve4ajWvRn6PXPMujpS9lip83qapcaBN93SarKoFUP2nVtQ9/DduSRK4eRtzCfvIX5PHLlMHY/ahfadmlT4zodN2tPwdJC/vePj4iiiHnTFzDqP1/Qo3d3AFYUFvHENcOZPuFHysvLWTRnCf964DV677ddXe7aRq992xx237kHDzz5Lkvzl7M0fzkPPPkue+26Oe3bZK913d+euhefj/2BsROqBq201GQO3X87HntmJHPn51G4vJghQ99ls65t6LVVpw25Ow3GtJfG0eOk3uT0aE1yegpbntOX8pIyZn8wtdZ15nw4lbwpi+hxcp9qy1pu24Euh2xFdrdWG7LZaqCOuKAfz9/zCtPGTad4RTFPXv8sKWkp7HnMrtXqrk9/fPj5B/Hpa2MY+eIoykrLGPniKEa/PobDL+hX17u2UfN9l6SqDFr1oEfvbhSvKGbK2B8qy6aM/YHiohJ69u5W4zqf/XcMMyfOYf9T9iQpKYmOm7Vnt8N3ZuRLo2p9nZ0O2oFJX9b+RbYp2rx7O4qKS5k0bX5l2aRp8ykuKaVn93a1rrfDNp3ZfecePPL0B9WWde3UimbpqUyYOLuybGn+cmbPW8oWm7WPdwcaoJJlRSyfk0/u1qvei6SUJHI2b0vexPk1rlO8dAXj/u99eg/an5BsN6X1l5mTScfu7fj200mVZeVl5Uz+cio9+nSvVn99+uMevbvx3eiJVdb7/rNJtfbXTZHvuyRV12ju0QohVB/PtUqHOmvIesjKyazxXqyCJQVk5mTUuE7R8mJee/xtBgw5h4FPDiA5JZk3/vYe/33i3RrrH3j63hx42t5cttf1sba9ocvKSKOghnuxlhUUkZWZXuM6Gc1SuXrAwdz+wOsUFZdWW56ZkQZAfkHV7eYvW0FWxbKmrLQgMTwwtXnV9yItO71y2ZrG3vseXQ/flpzNWrPgy5kbvI1at4bSx2ZV9KEFa9ybmr+4oHJZ1frr7o8zczJYtqRqnfzFBWTmZMbV7AbP973xWMex3mn27Nn8O/yd8qi8ztrUlCXNTqJz5xGJJ+XzgbJ6bU/TMp3Zc0sBaj8Tvw6NJmhtzPY/ZU9+//D5lc+HDHiMrBbV/6PIys2iMG95jdv4df99OPeOU7nh6LsYN/JbWrZvwWV//R2Dnr6EP53yf1XqHnregZzzp1O45pDbqpwtbIp+vffW/OF3B1U+v/fRt2oMVM2z0msMYAAXnbkvH38+ha++qfn/nsLlibCQnZXOwtWCWHbzZhQsX/s9SE1BSlYiYJUsq/peFOcX0axtVrX6M9+eSOHMPHa8/qBqy6R1KajoQ7Nyq362sltmsWDWohrqF66zPy7MW07z3Kp1sltmUZhX++RFTY3ve5MRlZeXly6nYF59N+RnWHlCaE69tuKnKoOZMxv0pGYN830HKkJtO6Dk526h0QStKIo617as4uxMvd0s887wkbwzfGTl83Zd25DWLI3uvboy9evpAHTv1ZW09FQmjZlW4za22LkHX78/ga8/mADAojlL+M+jb3Hts5dVqXfiwKM49vIjGPjrm5lcy7aakjffn8Cb70+ofN6+bQ7paSn02LQtk39IDFvrsWlb0lJTmDS15v83+vbpTvOsdH6999YANEtPJSU5iVf/dhHnX/V3ps9cxIqiErbq2YEPR08GoEV2Bh3atuD7KXM38B5u/FKbp5PRIZsl386l1XaJ/ra8tJy8SQvo3G/LavXnjZrOsumLeeOYoQBEpYmzpm8cM5RtL9yDLodsVXeNV6WNuY9dXWFeIbOnzmPLXXow4ZPvAUhKTmKz3t146+n3q9WfPGbaOvvjyWOmseXOPaust8VOPWrtr5si3/fGY23HekO28kpdY92/jVVTf9+9+aEezJu+gFEjvuD8wf3JaZ1NTutszh/cn49f/oz5MxbUuM7XH0yg195bs82vtgCgRZscDjnnACZ+PqWyzrl3nMrRFx/KlfveYMiqxdz5eXz02WQuOnNfWmRn0CI7g4vO3JeRn05i7oKap2I//6q/0//SoZx1+d846/K/8a//fsX3U+dx1uV/Y868pRSXlPGfd8Zx7sl70r5NNhnNUhlw1r5M+3EhX3/rsDeAbsdsx+Rnx5A3ZSFlRaV8N/RTklKS6LhX9Xs3tr14D/Z7+lT2efxE9nn8RHYYuB8Aez96PB337QFAVB5RVlRKeUliCEV5aTllRaVEZQ5lEbzy0H85/ooj6bZtF9KapXHGTSdSVlJW47Tg69Mfj/jrm+x6aB/2OHpXklOS2ePoXdnlkD68+vAbdb1rGzXfd0mqqtFc0Wpo7uw/hAFDzmHYpAcA+OTVzxky4LHK5SuHGx6ZczoA7//zY1p3bMkVj19I601aUry8mK/fn8Dtp90HJK6SnTjwaEqKS3ng0zuqvNY5215Wa4Brim657z9cft4BPPfQeQB89Nlk7v3rW5XLVw43POiUxHu7aI17DgoLiygtLWP+wmWVZUOeeJeLz96PJ+49g9SUZMaMn8FVt71IFNXBDjUAPU7qQ2lhCZ9c/m9KCkrI3bItfQcfQUpmGoVz83mv/3D63nUErXfYhLTsZrDaBJAFMxP3azRr07zyd7QWfjWLjy/9V2Wd9/onftOs99X70+WQretsv7Rx+ufdL5OZncGdb/6RzJwMvv9sMlcfchsrClbQtksbHh//Z6459DbGjfwWWHd/PGHURO7sP4Sz/3QK1wy/lDnT5nPH6ffz/WeT62X/Nla+75JUVYiawDfBEMKPnTp16rRXOIwFM6uPFdeG0aZTK56Z8QgAx5z7UJVgog2nbevmvPTYBQCcNeo6FhYvqd8GNSGt03J589gnmTlz5symNEzCPrZ+rN7HntzlfN/7OtKmUys+iEY0ueO8oWvqQ9jqS1N/3x06KEmSJEkxM2hJkiRJUswMWpIkSZIUMyfDkCRJUqPWVO8Rqm9N/X33ipYkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSJMXMoCVJkiRJMTNoSZIkSVLMDFqSJEmSFDODliRJkiTFzKAlSZIkSTEzaEmSJElSzAxakiRJkhQzg5YkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSVM9CCNNCCO/VdzsUH4OWJEmSmowQwo4hhHtCCF+GEJaEEBaGED4OIZwWQgj13T41HgYtSZIkNSUDgf7AKOAq4GagHHgKeKwe26VGJqW+GyBJkiTVofuBM6IoKlpZEEIYArwDnB1C+HMURePqrXVqNLyiJUmSpCYjiqKPVg9ZFWXlwAsVT7dbn+2EEFqGEO4KIUwMIRSFEOaHEN4OIfx6jXr7hBDeCiHkhRAKQwijQwgnr8f2u4UQohDCjTUsezKEEK1R9l7FfV7dQwgvV7zewhDC/SGE1BBCsxDCn0MIs0MIy0MIr4cQuqyxjRsrXnPLEMKdIYRZIYQVIYRRIYTd16ibFEK4IoTwdQhhWQhhaQhhQgjhgfV5/5oCr2hJkiRJ0Lnicf66KoYQWgMfAZsDzwL3AenAbsCBwJsV9Y4AXgJmAYOBAuA0YHgIoXMURYNj3ocs4G3gLRJDJA8ALgaKga2BZOBWYFPgMmAYsF8N2xlW0dY7gBzgCuDVEEL3KIqWVtS5DriJxJDLB0lcwOkBHBzzPjVYBi1JkiQ1aSGEjsBvgR+AD9ZjlT8BW5AYgjhsjW0lVTwmAw8AecCuURTNqSj/C4mQdmsI4ekoimbHtiPQBrg1iqL7Kp4/HEIYDVwOvBhF0XGrtTMAV4YQtoqi6Ns1tjMHODqKoqii7gTgeeBk4OGKOkcDr0VR1H+Nda+IcX8aNIcOSpIkqckKIaQD/yRx5eacKIqK11E/CTgRGLtmyILKYYgAOwFdgcdWhqyK5SuAe4A04LBYdmKVMlYFoZU+BALwlxrKIXFVbk1DVoasCu9WPPZcrWwpsG0IodfPbGujZ9CSJElSkxRCSAH+AewOnB9F0dvrsVpboAXw1Trqda94HF/DspVlm61PO3+C2WvefwYsqXj8oZbyVjVsZ+rqT6IoWlTxz9arFV8DZABjQwhTQgiPhxCOWXlFTwYtSZIkNUEVQ/uGA0cCl0ZR9FOndo/WXeUXWdv2k2spL1vLOrUtq+m3w9ZZN4qij0kExeOA14E9gReBj0IIGWtpR5Nh0JIkSVKTUnHV5SngeODKKIqG/ITV55O4GrTDOupNqXjcpoZl26xRpyYrryK1rGFZ3FfCfpYoipZFUfRCFEUXRlG0JYmrXH1JDK1s8gxakiRJajIqQtZQEhM7XBNF0T0/Zf2Ke7CeBXYIIZxaw/ZXXvX5AphO4re52q62PJ3E5BTFwIi1vE4+MJc1ZgUMIfQFfvVT2rwhhBDa1FD8ZcVj6xqWNTnOOihJkqSmZDDQHxgNzAghnLbG8rFRFI1dxzauJTF1+lMhhEOAj0lMbrEriXB1VRRFZSGEASSmdx8dQngUKCQxvfuOwMD1mHHwQeDmEMKrwKtAN+BcYCzrvqK2oU0IIXwMfEpi+vrOwAUkpoV/qT4btrEwaEmSJKkp2anicRcSwwfXdBOJIFOrKIoWhRB2I/FbUscAJ5AYTvgV8Nhq9V4JIRwIXA8MIvHdexxwahRFw9ejrXeQmKziVBLBbgzwG+Bs6j9o3UNi1sRLSczYOJfE7IS3RVG0tiGRTYZBS5IkSU1GFEX7xrSdRSSGAF6+jnrvAe+tx/a61VBWQuKHhS9bY9H7wJlr1N23lu3eCNxYS7vC+tStWLZm3TtIBEHVwnu0JEmSJClmBi1JkiRJiplBS5IkSZJiZtCSJEmSpJgZtCRJkiQpZgYtSZIkSYqZQUuSJEmSYhaiKKrvNsQihPDjWhZ3SkpKIj1kUF5WXmdtauqSkpNo1SEXgIWLCygvbxyftY1dUlKgdcssABYX51GOn/m6kkQSKxYWUF5eXhJFUVp9tydO9rEbn9X72EVzlvje15Gk5CSKouWN8jiXFK+mErQ2AUqBeXXUnLh1qHicU6+taHp83+tPQ37v2wElURRl1XdD4mQfqw2gIb/vjfI4lxSvRhO0GrOVX3CiKOpc321pSnzf64/vveqSn7f64fsuqbHzHi1JkiRJiplBS5IkSZJiZtCSJEmSpJgZtCRJkiQpZgYtSZIkSYqZQUuSJEmSYub07pIkSZIUM69oSZIkSVLMDFqSJEmSFDODliRJkiTFzKAlSZIkSTEzaEmSJElSzAxakiRJkhQzg5YkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSJMXMoCVJkiRJMTNoSZIkSVLMDFqSJEmSFDODliRJkiTFzKClJi+EMC2E8F59t0OSamM/JUkNj0FLdSaE0DWE8FQI4ZsQwpIQQmEI4dsQwp9DCB3ru32SFELYMYRwTwjhy4p+amEI4eMQwmkhhFDf7ZMkNRwp9d0ANSntga7Ay8AMoATYDjgPOCGE0DuKovn12D5JGggcALwAPAw0A04AngL2A86pv6ZJkhqSEEVRfbdBTVwI4XjgH8CVURTdUw+vPw2YFkXRvnX92pI2LiGE3YHPoygqWq0sCXgH2AfoFUXRuHpo1zTspySpQXHooDYGP1Q85q5P5RBCyxDCXSGEiSGEohDC/BDC2yGEX69Rb58QwlshhLyKYYqjQwgnr8f2u4UQohDCjTUsezKEEK1R9l7F/RPdQwgvV7zewhDC/SGE1BBCs4rhkbNDCMtDCK+HELqssY0bK15zyxDCnSGEWSGEFSGEURVf/FavmxRCuCKE8HUIYVkIYWkIYUII4YH1ef8k1S6Koo9WD1kVZeUkrnBB4ir8OtlP2U9JkkMHVedCCOlANpAObAPcUbHoP+uxbmvgI2Bz4Fngvort7AYcCLxZUe8I4CVgFjAYKABOA4aHEDpHUTQ4xl0CyALeBt5i1dCji4FiYGsgGbgV2BS4DBhGYhjSmoZVtPUOIAe4Ang1hNA9iqKlFXWuA24iMZTpQRInTHoAB8e8T5JW6VzxuM7hzfZTgP2UJBm0VC9OBoau9nwacGoURR+vx7p/ArYAzoiiaNjqCyqG9xBCSAYeAPKAXaMomlNR/hcSX35uDSE8HUXR7F+6I6tpA9waRdF9Fc8fDiGMBi4HXoyi6LjV2hmAK0MIW0VR9O0a25kDHB1VjOkNIUwAnifxnj1cUedo4LUoivqvse4VMe6PpAohMVnPb0lcff9gPVaxn7KfkiSHDqpe/Bf4NfAb4HYSZ0Zz17VSxReUE4Gxa355gcrhPQA7kZh047GVX14qlq8A7gHSgMN+2S5UU8aqLxgrfQgE4C81lEPibPeahqz88lLh3YrHnquVLQW2DSH0+pltlbSeKq7A/5PElZtzoigqXkd9+6kE+ylJTZ5XtFTnKs7QrjxL+1II4b/AeyGE4iiKHlvLqm2BFsBX63iJ7hWP42tYtrJss/Vt73qaveZ9HcCSiscfailvVcN2pq7+JIqiRYkTy7Rerfga4N/A2BDCVBJfcl4F/r3alzhJv1AIIYXERD27A7+Noujt9VjNfirBfkpSk+cVLdW7KIr+B/wInLW+q2zA5qxr+8m1lJetZZ3altX0mzzrrFsxxHIz4DjgdWBP4EXgoxBCxlraIWk9VQztGw4cCVy6jpNANbGfsp+S1MQZtLSxyABarqPOfBJnWXdYR70pFY/b1LBsmzXq1GRRxWNN7Yn7DPPPEkXRsiiKXoii6MIoirYkcfa4L4khS5J+gYrhf08Bx5P42YkhP2F1+6kK9lOSmjqDlupMCKF9LeUnkRhyMmpt61cMN3kW2CGEcGoN21l5NvULYDpwdgih7WrL00nc9F0MjFjL6+QDc1ljtq0QQl/gV2trY10IIbSpofjLisfWNSyTtJ4qQtZQEhM7XPNTf9vPfqqyHfZTkpo879FSXbozhLAd8AaJmQazSEx3fByJe7ZuXI9tXEtiSuKnQgiHAB+TuGl8VxJfWq6KoqgshDCAxLTJo0MIjwKFJKZN3hEYuB4zeT0I3BxCeJXEfQXdgHOBsaz7TPWGNiGE8DHwKYlpoTsDF5CYVOSl+myY1AgMBvoDo4EZIYTT1lg+NoqisevYhv2U/ZQkGbRUp14gMczldBI3jJeTCFx/Bu6KomjeujZQcdP1biR+o+UY4AQSw3S+Ah5brd4rIYQDgeuBQSQ+6+NITCM/fD3aegeJm8BPJfGFaQyJWRLPpv6/wNxDYjayS0nMhDaXxI3mt0VRtLahRpLWbaeKx11IDB9c000kgkyt7KcA+ylJIlSdoVWSJEmS9Et5j5YkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSJMXMoCVJkiRJMTNoSZIkSVLMDFqSJEmSFDODliRJkiTFzKAlSZIkSTEzaEmSJElSzAxakiRJkhQzg5YkSZIkxcygJUmSJEkxM2hJkiRJUswMWpIkSZIUM4OWJEmSJMXMoCVJkiRJMTNoSZIkSVLM/h8rLOjZmVB9xgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 840x840 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = pt.tensor(\n",
    "    [[1.0, 2.0],\n",
    "     [3.0, 4.0],\n",
    "     [5.0, 6.0]]\n",
    ")\n",
    "U, s, VH = pt.svd(A, some=False, compute_uv=True)\n",
    "S = pt.zeros_like(A)\n",
    "S[:A.shape[1], :] = pt.diag(s)\n",
    "\n",
    "assert pt.allclose(U.mm(U.conj().T), pt.eye(3), atol=1.0e-6)\n",
    "assert pt.allclose(VH.mm(VH.conj().T), pt.eye(2), atol=1.0e-6)\n",
    "assert pt.allclose(A, U.mm(S.mm(VH)))\n",
    "vis.plot_matrices_as_heatmap([U, S, VH], [r\"$U$\", r\"$\\Sigma$\", r\"$V^H$\"])"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}