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    "# Lecture 21: Matrices and Linear Algebra\n",
    "### Phys 503, J.R. Gladden, University of Mississippi"
   ]
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    "### Matrices in science\n",
    "Matrices are convenient mathematical constructs to represent various types of physical quantities.  A *vector* is an example of a 1D matrix and is used to represent a quantity that requires both a magnitude and direction.  Other quantities might require a magnitude, direction, and orientation - such as moment of intertia or elastic stiffness.  *Tensors* are used here and can be represented in a 2D matrix.  See the lecture slides for many more details.\n",
    "\n",
    "Matrices are also integral to representing and solving systems of linear equations - a field known in mathematics as linear algebra.\n"
   ]
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   "source": [
    "### Matrix objects in Numpy\n",
    "Numpy defines a \"matrix\" object that is similar to a 2D array, but adds certain capabilities for manipulating and combining matrices.\n"
   ]
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       "array([[ 1.,  0.,  0.],\n",
       "       [ 0.,  1.,  0.],\n",
       "       [ 0.,  0.,  1.]])"
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    "import numpy as np\n",
    "from numpy import linalg\n",
    "\n",
    "#2D numpy arrays can represent matrices\n",
    "\n",
    "Aa=np.array([[1,3,5],[2,5,1],[2,3,8]])\n",
    "\n",
    "#Numpy also has a 'matrix' object that is very similar to 2D arrays,\n",
    "#but has a more userfriendly interface\n",
    "\n",
    "A=np.matrix('[1 3 5; 2 5 1; 2 3 8]')\n",
    "\n",
    "#Returns the diagonal elements of the matrix\n",
    "\n",
    "A.diagonal()\n",
    "\n",
    "#Returns the trace (sum of diag. elements)\n",
    "A.trace()\n",
    "\n",
    "#Takes inverse of the matrix\n",
    "A.I\n",
    "\n",
    "# Matrix multiply using simple * operator\n",
    "# Should return identity matrix (matrix equivalent of 1)\n",
    "A.I*A\n",
    "\n",
    "#NxN Identity matrix given by eye(N), same syntax as Matlab\n",
    "np.eye(3)\n"
   ]
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     "data": {
      "text/plain": [
       "matrix([[1, 3, 5],\n",
       "        [2, 5, 1],\n",
       "        [2, 3, 8]])"
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   "source": [
    "A\n",
    "\n"
   ]
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   "source": [
    "### Solving systems of equations example\n",
    "$$ x + 3y + 5z = 10 \\\\ 2x + 5y + z = 8 \\\\ 2x + 3y + 8z = 3$$"
   ]
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     "text": [
      "[[-9.28]\n",
      " [ 5.16]\n",
      " [ 0.76]]\n"
     ]
    }
   ],
   "source": [
    "# Solutions to systems of equations\n",
    "# System\n",
    "#x + 3y + 5z = 10\n",
    "#2x + 5y + z = 8\n",
    "#2x + 3y + 8z = 3\n",
    "#\n",
    "#Form is A*X = b, \n",
    "\n",
    "b=np.matrix('[10; 8; 3]')\n",
    "A=np.matrix('[1 3 5; 2 5 1; 2 3 8]')\n",
    "\n",
    "# Solution can be found by 1/A * b\n",
    "A.I * b\n",
    "\n",
    "#Better to use the solve function in linalg\n",
    "X=linalg.solve(A,b)\n",
    "print(X)"
   ]
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       "matrix([[-9.28],\n",
       "        [ 5.16],\n",
       "        [ 0.76]])"
      ]
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    "A.I*b"
   ]
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   "source": [
    "### Eigenvalue Problems\n",
    "Eigenvalue (or \"proper\" value) problems are ones that can be cast in the form $$ \\matrix{A} \\matrix{X} = \\lambda \\matrix{X} $$\n",
    "Where $X$ is the eigenvector and $\\lambda$ are the eigenvalues.\n",
    "\n",
    "numpy.linalg provides tools for doing this."
   ]
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     "text": [
      "('Eigenvalues: ', array([ 10.5540456 ,  -0.5873064 ,   4.03326081]))\n",
      "Eigenvectors: \n",
      "[[-0.51686204 -0.94195144  0.11527992]\n",
      " [-0.32845853  0.31778071 -0.81936883]\n",
      " [-0.79054957  0.10836468  0.56155611]]\n"
     ]
    }
   ],
   "source": [
    "#Eigenvalues\n",
    "lambs,vs = linalg.eig(A)\n",
    "print(\"Eigenvalues: \",lambs)\n",
    "print(\"Eigenvectors: \")\n",
    "print(vs)"
   ]
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   "source": [
    "linalg.eig??\n"
   ]
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