5-Linear_algebra.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear algebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'1.14.3'"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.__version__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrix and vector products"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q1. Predict the results of the following code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[8 5]\n",
      "[6 6]\n",
      "[8 5]\n",
      "[6 6]\n",
      "[6 6]\n"
     ]
    }
   ],
   "source": [
    "x = [1,2]\n",
    "y = [[4, 1], [2, 2]]\n",
    "print(np.dot(x, y))\n",
    "print(np.dot(y, x))\n",
    "print(np.matmul(x, y))\n",
    "print(np.inner(x, y))\n",
    "print(np.inner(y, x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q2. Predict the results of the following code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[4 1]\n",
      " [2 2]\n",
      " [1 1]]\n",
      "[[4 1]\n",
      " [2 2]\n",
      " [1 1]]\n"
     ]
    }
   ],
   "source": [
    "x = [[1, 0], [0, 1]]\n",
    "y = [[4, 1], [2, 2], [1, 1]]\n",
    "print(np.dot(y, x))\n",
    "print(np.matmul(y, x))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q3. Predict the results of the following code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "30\n",
      "30\n",
      "30\n",
      "30\n",
      "30\n"
     ]
    }
   ],
   "source": [
    "x = np.array([[1, 4], [5, 6]])\n",
    "y = np.array([[4, 1], [2, 2]])\n",
    "print(np.vdot(x, y))\n",
    "print(np.vdot(y, x))\n",
    "print(np.dot(x.flatten(), y.flatten()))\n",
    "print(np.inner(x.flatten(), y.flatten()))\n",
    "print((x*y).sum())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q4. Predict the results of the following code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "abb\n",
      "abb\n",
      "[['a' 'aa']\n",
      " ['b' 'bb']]\n",
      "[['a' 'b']\n",
      " ['aa' 'bb']]\n"
     ]
    }
   ],
   "source": [
    "x = np.array(['a', 'b'], dtype=object)\n",
    "y = np.array([1, 2])\n",
    "print(np.inner(x, y))\n",
    "print(np.inner(y, x))\n",
    "print(np.outer(x, y))\n",
    "print(np.outer(y, x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Decompositions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q5. Get the lower-trianglular `L` in the Cholesky decomposition of x and verify it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.,  0.,  0.],\n",
       "       [ 6.,  1.,  0.],\n",
       "       [-8.,  5.,  3.]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array([[4, 12, -16], [12, 37, -43], [-16, -43, 98]], dtype=np.int32)\n",
    "np.linalg.cholesky(x)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q6. Compute the qr factorization of x and verify it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "q = [[-0.85714287  0.3942857   0.33142856]\n",
      " [-0.42857143 -0.9028571  -0.03428571]\n",
      " [ 0.2857143  -0.17142858  0.94285715]],\n",
      "\n",
      " r = [[ -14.  -21.   14.]\n",
      " [   0. -175.   70.]\n",
      " [   0.    0.  -35.]]\n"
     ]
    }
   ],
   "source": [
    "x = np.array([[12, -51, 4], [6, 167, -68], [-4, 24, -41]], dtype=np.float32)\n",
    "q,r = np.linalg.qr(x)\n",
    "print(\"q = {},\\n\\n r = {}\".format(q,r))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q7. Factor x by Singular Value Decomposition and verify it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u = [[ 0.  1.  0.  0.]\n",
      " [ 1.  0.  0.  0.]\n",
      " [ 0.  0.  0. -1.]\n",
      " [ 0.  0.  1.  0.]]\n",
      " s= [3.       2.236068 2.       0.      ]\n",
      " vh = [[-0.         0.         1.         0.         0.       ]\n",
      " [ 0.4472136  0.         0.         0.         0.8944272]\n",
      " [-0.         1.         0.         0.         0.       ]\n",
      " [ 0.         0.         0.         1.         0.       ]\n",
      " [-0.8944272  0.         0.         0.         0.4472136]]\n"
     ]
    }
   ],
   "source": [
    "x = np.array([[1, 0, 0, 0, 2], [0, 0, 3, 0, 0], [0, 0, 0, 0, 0], [0, 2, 0, 0, 0]], dtype=np.float32)\n",
    "u,s,vh = np.linalg.svd(x)\n",
    "print(\"u = {}\\n s= {}\\n vh = {}\".format(u,s,vh))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrix eigenvalues"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q8. Compute the eigenvalues and right eigenvectors of x. (Name them eigenvals and eigenvecs, respectively)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eigenvals = [1. 2. 3.] \n",
      "eigenvecs=\n",
      "[[1. 0. 0.]\n",
      " [0. 1. 0.]\n",
      " [0. 0. 1.]]\n"
     ]
    }
   ],
   "source": [
    "x = np.diag((1, 2, 3))\n",
    "eigenvals, eigenvecs = np.linalg.eig(x)\n",
    "print(\"eigenvals = {} \\neigenvecs=\\n{}\".format(eigenvals,eigenvecs))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q9. Predict the results of the following code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n"
     ]
    }
   ],
   "source": [
    "print(np.array_equal(np.dot(x, eigenvecs), eigenvals * eigenvecs))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Norms and other numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q10. Calculate the Frobenius norm and the condition number of x."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "16.881943016134134\n",
      "3.193239515623568e+17\n"
     ]
    }
   ],
   "source": [
    "x = np.arange(1, 10).reshape((3, 3))\n",
    "print(np.linalg.norm(x,ord='fro'))\n",
    "print(np.linalg.cond(x,p='fro'))\n",
    "# 16.8819430161\n",
    "# 4.56177073661e+17"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q11. Calculate the determinant of x."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.0000000000000004"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.arange(1, 5).reshape((2, 2))\n",
    "np.linalg.det(x) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q12. Calculate the rank of x."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.eye(4)\n",
    "np.linalg.matrix_rank(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q13. Compute the sign and natural logarithm of the determinant of x."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1.0, 0.6931471805599455)"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.arange(1, 5).reshape((2, 2))\n",
    "np.linalg.slogdet(x)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q14. Return the sum along the diagonal of x."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.0"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.eye(4)\n",
    "np.trace(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving equations and inverting matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q15. Compute the inverse of x."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "ufunc 'invert' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-85-b124f3f1816d>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[0mx\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0marray\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m1.\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m2.\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m[\u001b[0m\u001b[1;36m3.\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m4.\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      2\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 3\u001b[1;33m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0minvert\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m: ufunc 'invert' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''"
     ]
    }
   ],
   "source": [
    "x = np.array([[1., 2.], [3., 4.]])\n",
    "\n",
    "np.invert(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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