{
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   "cell_type": "markdown",
   "id": "73bd968b-d970-4a05-94ef-4e7abf990827",
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   "source": [
    "Chapter 06\n",
    "\n",
    "# 分块矩阵\n",
    "Book_4《矩阵力量》 | 鸢尾花书：从加减乘除到机器学习 (第二版)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25999c8b-f570-4ca4-b9d4-6e8c9537b859",
   "metadata": {},
   "source": [
    "这段代码演示了如何对矩阵进行分块和重组操作。首先定义了一个 \\(4 \\times 5\\) 矩阵 \\( A \\)：\n",
    "\n",
    "$$\n",
    "A = \\begin{bmatrix}\n",
    "1 & 2 & 3 & 0 & 0 \\\\\n",
    "4 & 5 & 6 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & -1 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 1 \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "然后使用 NumPy 的切片方法，将矩阵 \\( A \\) 分割为 4 个子矩阵（即 \\(2 \\times 2\\) 的块矩阵结构）：\n",
    "\n",
    "1. \\( A_{1,1} = A[0:2, 0:3] \\)，即前两行和前三列的子矩阵：\n",
    "   $$\n",
    "   A_{1,1} = \\begin{bmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \\end{bmatrix}\n",
    "   $$\n",
    "   \n",
    "2. \\( A_{1,2} = A[0:2, 3:] \\)，即前两行和最后两列的子矩阵：\n",
    "   $$\n",
    "   A_{1,2} = \\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\end{bmatrix}\n",
    "   $$\n",
    "\n",
    "3. \\( A_{2,1} = A[2:, 0:3] \\)，即后两行和前三列的子矩阵：\n",
    "   $$\n",
    "   A_{2,1} = \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix}\n",
    "   $$\n",
    "\n",
    "4. \\( A_{2,2} = A[2:, 3:] \\)，即后两行和最后两列的子矩阵：\n",
    "   $$\n",
    "   A_{2,2} = \\begin{bmatrix} -1 & 0 \\\\ 0 & 1 \\end{bmatrix}\n",
    "   $$\n",
    "\n",
    "最后使用 `np.block` 函数将这些子矩阵重新组合为一个新的矩阵 \\( A' \\)，其结构与原矩阵 \\( A \\) 相同：\n",
    "\n",
    "$$\n",
    "A' = \\begin{bmatrix} \n",
    "A_{1,1} & A_{1,2} \\\\ \n",
    "A_{2,1} & A_{2,2} \n",
    "\\end{bmatrix} = \n",
    "\\begin{bmatrix} \n",
    "1 & 2 & 3 & 0 & 0 \\\\ \n",
    "4 & 5 & 6 & 0 & 0 \\\\ \n",
    "0 & 0 & 0 & -1 & 0 \\\\ \n",
    "0 & 0 & 0 & 0 & 1 \n",
    "\\end{bmatrix}\n",
    "$$ \n",
    "\n",
    "通过这种方法，可以方便地在分块矩阵操作中对矩阵进行重新组合，实现了矩阵的分解与重构。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "da9bd893-e25f-4193-b12b-e5cac674f64b",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "67b49f8e-b6d5-49d4-b4a4-190ecd915450",
   "metadata": {},
   "source": [
    "## 定义矩阵 A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "e00b27dd-c721-42e1-8f65-fa2937b460ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1,  2,  3,  0,  0],\n",
       "       [ 4,  5,  6,  0,  0],\n",
       "       [ 0,  0,  0, -1,  0],\n",
       "       [ 0,  0,  0,  0,  1]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[1, 2, 3, 0,  0],   # 定义矩阵 A 的元素\n",
    "              [4, 5, 6, 0,  0],\n",
    "              [0, 0, 0, -1, 0],\n",
    "              [0, 0 ,0, 0,  1]])\n",
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5a878c1b-238b-4f8d-a033-db66fb2aa0fa",
   "metadata": {},
   "source": [
    "## NumPy 数组切片操作"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "440560e5-ed77-4bbc-8fa7-d2c09285e3c7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3],\n",
       "       [4, 5, 6]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_1_1 = A[0:2,0:3]  # 提取矩阵 A 的前两行和前三列作为子矩阵 A_1_1\n",
    "A_1_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "29344301-cbb7-4fbd-b64b-d23aced6823e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 0],\n",
       "       [0, 0]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_1_2 = A[0:2,3:]  # 提取矩阵 A 的前两行和最后两列作为子矩阵 A_1_2\n",
    "A_1_2\n",
    "# A_1_2 = A[0:2,-2:]  # 或者用负索引方式提取前两行和最后两列（注释部分）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "b24013dd-a04e-44f6-9c89-67b2a0027bff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 0, 0],\n",
       "       [0, 0, 0]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_2_1 = A[2:,0:3]  # 提取矩阵 A 的后两行和前三列作为子矩阵 A_2_1\n",
    "A_2_1\n",
    "# A_2_1 = A[-2:,0:3]  # 或者用负索引方式提取后两行和前三列（注释部分）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "2aa41d01-ef5a-450d-b912-d0cf9134f16f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-1,  0],\n",
       "       [ 0,  1]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_2_2 = A[2:,3:]  # 提取矩阵 A 的后两行和最后两列作为子矩阵 A_2_2\n",
    "A_2_2\n",
    "# A_2_2 = A[-2:,-2:]  # 或者用负索引方式提取后两行和最后两列（注释部分）"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f618ccce-d822-417d-8ce1-6cb6d3f90999",
   "metadata": {},
   "source": [
    "## 使用嵌套列表中的块组装矩阵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "4dc81f48-c1a3-42e0-82df-b76c62601926",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1,  2,  3,  0,  0],\n",
       "       [ 4,  5,  6,  0,  0],\n",
       "       [ 0,  0,  0, -1,  0],\n",
       "       [ 0,  0,  0,  0,  1]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_ = np.block([[A_1_1, A_1_2],   # 通过 np.block 函数将子矩阵 A_1_1、A_1_2、A_2_1 和 A_2_2 组合成新的矩阵 A_\n",
    "               [A_2_1, A_2_2]])\n",
    "A_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "85a80909-2aac-49ed-bb7a-f8cc6b80ee7d",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ecd322f4-f919-4be2-adc3-69d28ef25e69",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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