{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "bbb1a6ac",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1f0556e",
   "metadata": {},
   "source": [
    "### Wyznacznik macierzy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f3ddf67e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[2, -1, 3], [0, 1, 1], [1, -1, 2]])\n",
    "np.linalg.det(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "29304216",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[1, 3, 2], [2, -1, 4], [-1, 2, -2]])\n",
    "np.linalg.det(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "168d5afa",
   "metadata": {},
   "source": [
    "### Macierz transponowana"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ef31857e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1,  2, -1],\n",
       "       [ 3, -1,  2],\n",
       "       [ 2,  4, -2]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.transpose(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "091ff847",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.det(np.transpose(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a7eda4d",
   "metadata": {},
   "source": [
    "### Macierz odwrotna"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "33c17224",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3., -1.],\n",
       "       [-5.,  2.]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[2, 1], [5, 3]])\n",
    "np.linalg.inv(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "6a21bc66",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  1.,  -1.,   1.],\n",
       "       [-38.,  41., -34.],\n",
       "       [ 27., -29.,  24.]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[2, 5, 7], [6, 3, 4], [5, -2, -3]])\n",
    "np.linalg.inv(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ec1ea19",
   "metadata": {},
   "source": [
    "### Zastosowanie wyznacznika do rozwiązywania układu 3 równań liniowych z 3 niewiadomymi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "631c89fd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2., -1., -1.])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[3, 4, 1], [2, -3, 2], [1, -1, 0]])\n",
    "B = np.array([1, 5, 3])\n",
    "np.linalg.solve(A, B)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4364bb47",
   "metadata": {},
   "source": [
    "### Działanie macierzy na wektor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "0981a527",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2, 13,  9])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[2, -1, 1], [4, 2, 5], [2, 1, 0]])\n",
    "B = np.array([2, 5, -1])\n",
    "np.dot(A, B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "fec9feda",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2],\n",
       "       [13],\n",
       "       [ 9]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[2, -1, 1], [4, 2, 5], [2, 1, 0]])\n",
    "B = np.array([[2], [5], [-1]])\n",
    "np.dot(A, B)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65b3007b",
   "metadata": {},
   "source": [
    "### Mnożenie macierzy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "d1b4f8d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 21, -13,  -9],\n",
       "       [ 52,  22,  49],\n",
       "       [ 14,  -6,  -2]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = np.array([[3, 0, 8], [7, 1, -5], [2, 0, 4]])\n",
    "D = np.array([[7, 1, 5], [3, 5, -1], [0, -2, -3]])\n",
    "np.dot(C, D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "c8ecd580",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 21, -13,  -9],\n",
       "       [ 52,  22,  49],\n",
       "       [ 14,  -6,  -2]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.matmul(C, D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "abd5b934",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-292],\n",
       "       [ 623],\n",
       "       [-124]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.multi_dot([C, D, A, B])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "b2947f4d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "AxB = \n",
      " [[  3 -11 -12]\n",
      " [  7  -1  12]\n",
      " [ -7  -1  -4]]\n",
      "BxA = \n",
      " [[ -1  -1 -12]\n",
      " [  3   9   2]\n",
      " [  6  -4 -10]]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[3, 2, -1], [0, 1, 5], [1, -2, -3]])\n",
    "B = np.array([[0, -3, -1], [2, -1, -3], [1, 0, 3]])\n",
    "print('AxB = \\n', np.dot(A, B))\n",
    "print('BxA = \\n', np.dot(B, A))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1b7634a3",
   "metadata": {},
   "source": [
    "### Wartości i wektory własne"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "7b05abf3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([1., 3.]),\n",
       " array([[1.        , 0.92847669],\n",
       "        [0.        , 0.37139068]]))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[1, 5], [0, 3]])\n",
    "np.linalg.eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "3ad28718",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-1.,  1.,  2.]),\n",
       " array([[ 0.        ,  0.70710678,  0.66666667],\n",
       "        [ 0.        ,  0.        ,  0.33333333],\n",
       "        [ 1.        , -0.70710678, -0.66666667]]))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[1, 2, 0], [0, 2, 0], [-2, -2, -1]])\n",
    "np.linalg.eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "329bb856",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( -1, \\  1, \\  \\left[ \\left[\\begin{matrix}0\\\\0\\\\1\\end{matrix}\\right]\\right]\\right), \\  \\left( 1, \\  1, \\  \\left[ \\left[\\begin{matrix}-1\\\\0\\\\1\\end{matrix}\\right]\\right]\\right), \\  \\left( 2, \\  1, \\  \\left[ \\left[\\begin{matrix}-1\\\\- \\frac{1}{2}\\\\1\\end{matrix}\\right]\\right]\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛       ⎡⎡0⎤⎤⎞  ⎛      ⎡⎡-1⎤⎤⎞  ⎛      ⎡⎡ -1 ⎤⎤⎞⎤\n",
       "⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢  ⎥⎥⎟  ⎜      ⎢⎢    ⎥⎥⎟⎥\n",
       "⎢⎜-1, 1, ⎢⎢0⎥⎥⎟, ⎜1, 1, ⎢⎢0 ⎥⎥⎟, ⎜2, 1, ⎢⎢-1/2⎥⎥⎟⎥\n",
       "⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢  ⎥⎥⎟  ⎜      ⎢⎢    ⎥⎥⎟⎥\n",
       "⎣⎝       ⎣⎣1⎦⎦⎠  ⎝      ⎣⎣1 ⎦⎦⎠  ⎝      ⎣⎣ 1  ⎦⎦⎠⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sp\n",
    "sp.init_printing(use_unicode=True)\n",
    "\n",
    "A = sp.Matrix([[1, 2, 0], [0, 2, 0], [-2, -2, -1]])\n",
    "A.eigenvects()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "ede40a5b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 4.,  1., -2.]),\n",
       " array([[-0.66666667, -0.66666667,  0.33333333],\n",
       "        [ 0.66666667, -0.33333333,  0.66666667],\n",
       "        [-0.33333333,  0.66666667,  0.66666667]]))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[2, -2, 0], [-2, 1, -2], [0, -2, 0]])\n",
    "np.linalg.eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "c601c4dc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( -2, \\  1, \\  \\left[ \\left[\\begin{matrix}\\frac{1}{2}\\\\1\\\\1\\end{matrix}\\right]\\right]\\right), \\  \\left( 1, \\  1, \\  \\left[ \\left[\\begin{matrix}-1\\\\- \\frac{1}{2}\\\\1\\end{matrix}\\right]\\right]\\right), \\  \\left( 4, \\  1, \\  \\left[ \\left[\\begin{matrix}2\\\\-2\\\\1\\end{matrix}\\right]\\right]\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛       ⎡⎡1/2⎤⎤⎞  ⎛      ⎡⎡ -1 ⎤⎤⎞  ⎛      ⎡⎡2 ⎤⎤⎞⎤\n",
       "⎢⎜       ⎢⎢   ⎥⎥⎟  ⎜      ⎢⎢    ⎥⎥⎟  ⎜      ⎢⎢  ⎥⎥⎟⎥\n",
       "⎢⎜-2, 1, ⎢⎢ 1 ⎥⎥⎟, ⎜1, 1, ⎢⎢-1/2⎥⎥⎟, ⎜4, 1, ⎢⎢-2⎥⎥⎟⎥\n",
       "⎢⎜       ⎢⎢   ⎥⎥⎟  ⎜      ⎢⎢    ⎥⎥⎟  ⎜      ⎢⎢  ⎥⎥⎟⎥\n",
       "⎣⎝       ⎣⎣ 1 ⎦⎦⎠  ⎝      ⎣⎣ 1  ⎦⎦⎠  ⎝      ⎣⎣1 ⎦⎦⎠⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sp.Matrix([[2, -2, 0], [-2, 1, -2], [0, -2, 0]])\n",
    "A.eigenvects()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "218a2257",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.11.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}