Basic Numerical Linear Algebra/11-The_Determinant.ipynb

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      "name": "python3",
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    "language_info": {
      "name": "python"
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
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      "source": [
        "<a href=\"https://colab.research.google.com/github/MonitSharma/Numerical-Linear-Algebra/blob/main/The_Determinant.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "KQ3k0blRyVS_"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import seaborn as sns"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Plot style\n",
        "sns.set()\n",
        "%pylab inline\n",
        "pylab.rcParams['figure.figsize'] = (4, 4)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "fbtQDH0sycfn",
        "outputId": "71f00911-5d54-4288-efe9-439c513dd3d4"
      },
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Populating the interactive namespace from numpy and matplotlib\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "def plotVectors(vecs, cols, alpha=1):\n",
        "    \"\"\"\n",
        "    Plot set of vectors.\n",
        "\n",
        "    Parameters\n",
        "    ----------\n",
        "    vecs : array-like\n",
        "        Coordinates of the vectors to plot. Each vectors is in an array. For\n",
        "        instance: [[1, 3], [2, 2]] can be used to plot 2 vectors.\n",
        "    cols : array-like\n",
        "        Colors of the vectors. For instance: ['red', 'blue'] will display the\n",
        "        first vector in red and the second in blue.\n",
        "    alpha : float\n",
        "        Opacity of vectors\n",
        "\n",
        "    Returns:\n",
        "\n",
        "    fig : instance of matplotlib.figure.Figure\n",
        "        The figure of the vectors\n",
        "    \"\"\"\n",
        "    plt.axvline(x=0, color='#A9A9A9', zorder=0)\n",
        "    plt.axhline(y=0, color='#A9A9A9', zorder=0)\n",
        "\n",
        "    for i in range(len(vecs)):\n",
        "        if (isinstance(alpha, list)):\n",
        "            alpha_i = alpha[i]\n",
        "        else:\n",
        "            alpha_i = alpha\n",
        "        x = np.concatenate([[0,0],vecs[i]])\n",
        "        plt.quiver([x[0]],\n",
        "                   [x[1]],\n",
        "                   [x[2]],\n",
        "                   [x[3]],\n",
        "                   angles='xy', scale_units='xy', scale=1, color=cols[i],\n",
        "                  alpha=alpha_i)"
      ],
      "metadata": {
        "id": "3iOqi4aWyfKQ"
      },
      "execution_count": 3,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "#Introduction\n",
        "This chapter is also very light! We will see what is the meaning of the determinant of a matrix. This special number can tell us a lot of things about our matrix!"
      ],
      "metadata": {
        "id": "gxW7pCTBymq-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# The Determinant\n",
        "We saw previously that a matrix can be seen as a linear transformation of the space. The determinant of a matrix $A$\n",
        " is a number corresponding to the multiplicative change you get when you transform your space with this matrix. \n",
        " \n",
        " -------\n",
        " \n",
        " A negative determinant means that there is a change in orientation (and not just a rescaling and/or a rotation). A change in orientation means for instance in 2D that we take a plane out of these 2 dimensions, do some transformations and get back to the initial 2D space.\n",
        "\n",
        "\n",
        " -------\n",
        "\n",
        " *The determinant of a matrix can tell you a lot of things about the transformation associated with this matrix*\n",
        "\n",
        " "
      ],
      "metadata": {
        "id": "O2rNbNNtyr5u"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Example 1.\n",
        "To calculate the area of the shapes, we will use simple squares in 2 dimensions. The unit square area can be calculated with the Pythagorean theorem taking the two unit vectors.\n",
        "\n",
        "Let's start by creating both vectors in Python:"
      ],
      "metadata": {
        "id": "PvSw2wWYzJBx"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "orange = '#FF9A13'\n",
        "blue = '#1190FF'\n",
        "    \n",
        "i = [0, 1]\n",
        "j = [1, 0]\n",
        "\n",
        "plotVectors([i, j], [[blue], [orange]])\n",
        "plt.xlim(-0.5, 3)\n",
        "plt.ylim(-0.5, 3)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 272
        },
        "id": "yUMRHyaAykOG",
        "outputId": "bd97571b-c013-4bf7-f1da-4382afb60690"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We will apply\n",
        "\n",
        "$${A}=\\begin{bmatrix}\n",
        "    2 & 0\\\\\\\\\n",
        "    0 & 2\n",
        "\\end{bmatrix}$$\n",
        "\n",
        "to $i$\n",
        " and $j$\n",
        ". You can notice that this matrix is special: it is diagonal. So it will only rescale our space. No rotation here. More precisely, it will rescale each dimension the same way because the diagonal values are identical. Let's create the matrix $A$\n",
        ":"
      ],
      "metadata": {
        "id": "4DkrbZ04zTzm"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "A = np.array([[2, 0], [0, 2]])\n",
        "A"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "lnY3WnufzQxs",
        "outputId": "91624d52-61f5-4f29-c814-d953b2d4c9ca"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[2, 0],\n",
              "       [0, 2]])"
            ]
          },
          "metadata": {},
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now we will apply $A$\n",
        " on our two unit vectors \n",
        " $i$ and $j$\n",
        " and plot the resulting new vectors:"
      ],
      "metadata": {
        "id": "AUHdGoE1zjQa"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "new_i = A.dot(i)\n",
        "new_j = A.dot(j)\n",
        "plotVectors([new_i, new_j], [['#1190FF'], ['#FF9A13']])\n",
        "plt.xlim(-0.5, 3)\n",
        "plt.ylim(-0.5, 3)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 272
        },
        "id": "FxRsJwZUzhCU",
        "outputId": "180c5167-df18-4dcf-e02c-6cd81e376274"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "As expected, we can see that the square corresponding to \n",
        " $i$ and $j$\n",
        " didn't rotate but the lengths of \n",
        "$i$ and $j$\n",
        " have doubled. We will now calculate the determinant of $A$\n",
        "\n",
        "\n",
        "\n",
        " -----\n",
        "\n",
        "\n",
        " *The unit square transformed by the matrix*"
      ],
      "metadata": {
        "id": "9ddtxfmkztuP"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "np.linalg.det(A)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "OtRij2ORzp6S",
        "outputId": "9d3611cd-7d44-4c0f-d106-fc954122cdb0"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "4.0"
            ]
          },
          "metadata": {},
          "execution_count": 7
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Example 2.\n",
        "Let's see now an example of negative determinant.\n",
        "\n",
        "We will transform the unit square with the matrix:\n",
        "\n",
        "$$ {B}=\\begin{bmatrix}\n",
        "    -2 & 0\\\\\\\\\n",
        "    0 & 2\n",
        "\\end{bmatrix}$$\n",
        "\n",
        "its determinant is $-4$"
      ],
      "metadata": {
        "id": "doUEqZtO0DNw"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "B = np.array([[-2, 0], [0, 2]])\n",
        "np.linalg.det(B)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "VoNX8Spcz9wx",
        "outputId": "cf606c8c-1242-462c-b4f2-001883cc03ce"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "-4.0"
            ]
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "new_i_1 = B.dot(i)\n",
        "new_j_1 = B.dot(j)\n",
        "plotVectors([new_i_1, new_j_1], [['#1190FF'], ['#FF9A13']])\n",
        "plt.xlim(-3, 0.5)\n",
        "plt.ylim(-0.5, 3)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 272
        },
        "id": "5zEc5Ax50Qs8",
        "outputId": "b81a6b1c-d978-4be7-ec1c-5ea4eccf3f2c"
      },
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We can see that the matrices with determinant $2$\n",
        " and $-2$\n",
        " modified the area of the unit square the same way."
      ],
      "metadata": {
        "id": "Yz2kHpC00W6K"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "The absolute value of the determinant shows that, as in the first example, the area of the new square is 4 times the area of the unit square. But this time, it was not just a rescaling but also a transformation. It is not obvious with only the unit vectors so let's transform some random points. We will use the matrix\n",
        "\n",
        "$${C}=\\begin{bmatrix}\n",
        "    -1 & 0\\\\\\\\\n",
        "    0 & 1\n",
        "\\end{bmatrix}$$\n",
        "\n",
        "that has a determinant equal to \n",
        " $-1$ for simplicity"
      ],
      "metadata": {
        "id": "T1wbaXeV0drz"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Some random points\n",
        "points = np.array([[1, 3], [2, 2], [3, 1], [4, 7], [5, 4]])"
      ],
      "metadata": {
        "id": "HAa0EVmQ0TRG"
      },
      "execution_count": 10,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "C = np.array([[-1, 0], [0, 1]])\n",
        "np.linalg.det(C)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Hc7nn8ao0sKp",
        "outputId": "1af1900a-93d3-4a7b-f06f-01a3ff97682e"
      },
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "-1.0"
            ]
          },
          "metadata": {},
          "execution_count": 11
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Since the determinant is $-1$\n",
        ", the area of the space will not be changed. However, since it is negative we will observe a transformation that we can't obtain through rotation:"
      ],
      "metadata": {
        "id": "BlrYSxFt0xul"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "newPoints = points.dot(C)\n",
        "\n",
        "plt.figure()\n",
        "plt.plot(points[:, 0], points[:, 1])\n",
        "plt.plot(newPoints[:, 0], newPoints[:, 1])\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 268
        },
        "id": "oDl3BphQ0u6z",
        "outputId": "fe3253b8-fc1d-461a-a070-b744b714bad9"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "You can see that the transformation mirrored the initial shape.\n",
        "\n",
        "## Conclusion\n",
        "We have seen that the determinant of a matrix is a special value telling us a lot of things on the transformation corresponding to this matrix. Now hang on and go to the last chapter on the Principal Component Analysis (PCA)."
      ],
      "metadata": {
        "id": "lID0vsBq1NmL"
      }
    }
  ]
}