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https://github.com/Visualize-ML/Book4_Power-of-Matrix.git
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363 lines
8.5 KiB
Plaintext
363 lines
8.5 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "73bd968b-d970-4a05-94ef-4e7abf990827",
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"metadata": {},
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"source": [
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"Chapter 04\n",
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"\n",
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"# 广播机制\n",
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"Book_4《矩阵力量》 | 鸢尾花书：从加减乘除到机器学习 (第二版)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "61a42cc1-9923-4983-973d-b03b59d6acc1",
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"metadata": {},
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"source": [
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"这段代码展示了 NumPy 的广播机制，通过定义矩阵、标量、列向量和行向量进行一系列加法操作。代码首先定义了一个 $3 \\times 2$ 的矩阵 $A$、一个标量 $k$、一个列向量 $c$ 和一个行向量 $r$。它们的具体定义如下：\n",
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"\n",
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"$$\n",
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"A = \\begin{bmatrix} 1 & 2 \\\\ 3 & 4 \\\\ 5 & 6 \\end{bmatrix}, \\quad k = 2, \\quad c = \\begin{bmatrix} 3 \\\\ 2 \\\\ 1 \\end{bmatrix}, \\quad r = \\begin{bmatrix} 2 & 1 \\end{bmatrix}\n",
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"$$\n",
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"\n",
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"代码使用 NumPy 广播机制，分别计算以下几种加法操作：\n",
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"\n",
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"1. **矩阵 $A$ 与标量 $k$ 相加**：标量 $k$ 会广播到矩阵 $A$ 的每个元素，即每个元素都加上 $k$ 值。这种情况下，运算结果为：\n",
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"\n",
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" $$\n",
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" A + k = \\begin{bmatrix} 1+2 & 2+2 \\\\ 3+2 & 4+2 \\\\ 5+2 & 6+2 \\end{bmatrix} = \\begin{bmatrix} 3 & 4 \\\\ 5 & 6 \\\\ 7 & 8 \\end{bmatrix}\n",
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" $$\n",
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"\n",
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"2. **矩阵 $A$ 与列向量 $c$ 相加**：列向量 $c$ 被广播到矩阵 $A$ 的每一列，使得 $c$ 的每个元素与 $A$ 相同行的每个元素相加，得到：\n",
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"\n",
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" $$\n",
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" A + c = \\begin{bmatrix} 1+3 & 2+3 \\\\ 3+2 & 4+2 \\\\ 5+1 & 6+1 \\end{bmatrix} = \\begin{bmatrix} 4 & 5 \\\\ 5 & 6 \\\\ 6 & 7 \\end{bmatrix}\n",
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" $$\n",
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"\n",
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"3. **矩阵 $A$ 与行向量 $r$ 相加**：行向量 $r$ 被广播到矩阵 $A$ 的每一行，使得 $r$ 的每个元素与 $A$ 相同行的每个元素相加，结果为：\n",
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"\n",
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" $$\n",
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" A + r = \\begin{bmatrix} 1+2 & 2+1 \\\\ 3+2 & 4+1 \\\\ 5+2 & 6+1 \\end{bmatrix} = \\begin{bmatrix} 3 & 3 \\\\ 5 & 5 \\\\ 7 & 7 \\end{bmatrix}\n",
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" $$\n",
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"\n",
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"4. **列向量 $c$ 与行向量 $r$ 相加**：通过广播，$c$ 和 $r$ 组合成一个 $3 \\times 2$ 的矩阵。每个 $c$ 的元素加到每一行的 $r$ 上，得到：\n",
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"\n",
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" $$\n",
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" c + r = \\begin{bmatrix} 3+2 & 3+1 \\\\ 2+2 & 2+1 \\\\ 1+2 & 1+1 \\end{bmatrix} = \\begin{bmatrix} 5 & 4 \\\\ 4 & 3 \\\\ 3 & 2 \\end{bmatrix}\n",
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" $$\n",
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"\n",
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"该代码展示了 NumPy 广播机制的灵活性，使得矩阵、向量和标量间的加法运算在不同维度下可以直接进行。"
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]
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},
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{
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"cell_type": "markdown",
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"id": "dbbb9e8f-480a-4b84-be1c-10299faa79c3",
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"metadata": {},
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"source": [
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"## 导入所需库"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "2da2788b-62a1-4402-bcc8-7c4448392cb3",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np # 导入NumPy库，用于数值计算"
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]
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},
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{
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"cell_type": "markdown",
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"id": "371997be-719d-416d-9e23-6c0c9ae9bc96",
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"metadata": {},
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"source": [
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"## 定义矩阵"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "7acefee2-ddb0-4ec5-924e-b15c468f3bda",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"matrix([[1, 2],\n",
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" [3, 4],\n",
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" [5, 6]])"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A = np.matrix([[1, 2], # 定义矩阵A\n",
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" [3, 4],\n",
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" [5, 6]])\n",
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"A"
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]
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},
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{
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"cell_type": "markdown",
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"id": "6a695575-9a76-41e6-bb17-6d77d1dbca49",
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"metadata": {},
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"source": [
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"## 定义标量"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "7b44157d-b5c4-450c-b5a9-8c17ec1fb9fc",
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"metadata": {},
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"outputs": [],
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"source": [
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"k = 2 # 定义标量k"
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]
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},
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{
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"cell_type": "markdown",
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"id": "950972dd-22ff-4a81-9793-2172ba4cc752",
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"metadata": {},
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"source": [
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"## 定义列向量"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "f59ea6bd-d607-48cf-8a7f-80eb5aac438c",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[3],\n",
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" [2],\n",
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" [1]])"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"c = np.array([[3], # 定义列向量c\n",
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" [2],\n",
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" [1]])\n",
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"c"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1a4a65e0-ce82-414a-990b-64713bca3913",
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"metadata": {},
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"source": [
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"## 定义行向量"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "c571bb39-d6b7-44ea-b5d7-2b197223eb5f",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[2, 1]])"
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]
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"r = np.array([[2, 1]]) # 定义行向量r\n",
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"r"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7a7d2094-3824-4efa-a099-40781551d1ff",
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"metadata": {},
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"source": [
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"## 广播原则"
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]
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},
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{
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"cell_type": "markdown",
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"id": "82e777a0-142f-4a7d-b3e1-2e38af1fe399",
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"metadata": {},
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"source": [
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"## 矩阵A加标量k"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "c7c158d0-9576-471c-953f-8767f19d9473",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"matrix([[3, 4],\n",
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" [5, 6],\n",
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" [7, 8]])"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A_plus_k = A + k # 矩阵A与标量k相加\n",
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"A_plus_k"
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]
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},
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{
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"cell_type": "markdown",
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"id": "07c8aa0c-d857-4267-a1b7-99c7fc856f6d",
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"metadata": {},
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"source": [
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"## 矩阵A加列向量c"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "919ef4b1-d43d-4c1e-933c-f9c8fde768f1",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"matrix([[4, 5],\n",
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" [5, 6],\n",
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" [6, 7]])"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A_plus_c = A + c # 矩阵A与列向量c相加\n",
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"A_plus_c"
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]
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},
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{
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"cell_type": "markdown",
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"id": "07bd091a-0642-400a-8fd2-57ea4588875a",
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"metadata": {},
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"source": [
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"## 矩阵A加行向量r"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "3a18f414-460e-452e-808e-47fd9d339345",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"matrix([[3, 3],\n",
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" [5, 5],\n",
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" [7, 7]])"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A_plus_r = A + r # 矩阵A与行向量r相加\n",
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"A_plus_r"
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]
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},
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{
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"cell_type": "markdown",
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"id": "5fd72975-156a-46c3-abce-170467f5d416",
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"metadata": {},
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"source": [
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"## 列向量c加行向量r"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"id": "e5ad12d3-b203-4a4e-90c4-20b693dfa4a2",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[5, 4],\n",
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" [4, 3],\n",
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" [3, 2]])"
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]
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},
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"execution_count": 9,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"c_plus_r = c + r # 列向量c与行向量r相加\n",
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"c_plus_r"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "85a80909-2aac-49ed-bb7a-f8cc6b80ee7d",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "ecd322f4-f919-4be2-adc3-69d28ef25e69",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.12.7"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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