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Book4_Ch14_Python_Codes/Bk4_Ch14_01.ipynb
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242
Book4_Ch14_Python_Codes/Bk4_Ch14_01.ipynb
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|
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{
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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 14\n",
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"\n",
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"# 矩阵平方根\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": "c263fc95-881a-49fb-9c6c-a25508996623",
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"metadata": {},
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||||
"source": [
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"该代码的主要任务是通过矩阵分解和重构方法，基于给定的矩阵$A$，验证矩阵重构的正确性。具体流程如下：\n",
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"\n",
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"1. **定义矩阵$A$**： \n",
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" 代码首先创建了一个$2 \\times 2$矩阵 $A = \\begin{bmatrix} 1.25 & -0.75 \\\\ -0.75 & 1.25 \\end{bmatrix}$。矩阵 $A$ 是一个对称矩阵，因此可以进行特征值分解。\n",
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"\n",
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"2. **计算特征值和特征向量**： \n",
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" 代码使用 `np.linalg.eig` 函数对矩阵 $A$ 进行特征值分解，计算出$A$的特征值（存储在 $LAMBDA$ 中）和特征向量（存储在 $V$ 中），满足以下分解公式：\n",
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" $$\n",
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" A = V \\Lambda V^{-1}\n",
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" $$\n",
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" 其中，$\\Lambda$ 是一个包含特征值的对角矩阵，$V$ 是由特征向量组成的矩阵。\n",
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"\n",
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"3. **构建矩阵$B$**： \n",
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" 接下来，代码构建了一个新的矩阵 $B$。它的计算公式为：\n",
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" $$\n",
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" B = V \\sqrt{\\Lambda} V^{-1}\n",
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" $$\n",
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" 其中，$\\sqrt{\\Lambda}$ 是对角矩阵，其对角元素是 $\\Lambda$ 的平方根。也就是说，$B$ 是通过将 $A$ 的特征值取平方根后重新组合得到的矩阵。\n",
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"\n",
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"4. **重构矩阵$A$并验证结果**： \n",
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" 最后，通过矩阵 $B$ 构造了一个新矩阵 $A_{reproduced}$，其计算公式为：\n",
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" $$\n",
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" A_{reproduced} = B B^T\n",
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" $$\n",
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" 这是利用矩阵 $B$ 重构 $A$ 的过程。对称矩阵 $A$ 的特征值平方根分解使得 $B B^T$ 应等于原始矩阵 $A$。因此，通过打印 $A_{reproduced}$，可以验证 $B B^T$ 是否等于 $A$，从而确认重构的正确性。"
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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": "2759881c-9e2a-4e4d-a79c-1617f2a4be4f",
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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"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a685835a-bcda-40f8-ac57-ff3738c0209f",
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"metadata": {},
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"source": [
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"## 初始化矩阵A"
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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": "a6e9a15c-1d77-4a95-a13c-1c825598e8be",
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"metadata": {},
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"outputs": [],
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"source": [
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"A = np.matrix([[1.25, -0.75], # 定义矩阵A\n",
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" [-0.75, 1.25]]) # 矩阵的元素"
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]
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},
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{
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"cell_type": "markdown",
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"id": "55297e89-757e-4136-a0af-4763d1db3e56",
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"metadata": {},
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"source": [
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"## 计算特征值和特征向量"
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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": 3,
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"id": "998fa920-5e90-408f-8b70-dd3633c870e9",
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"metadata": {},
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"outputs": [],
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"source": [
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"LAMBDA, V = np.linalg.eig(A) # 计算矩阵A的特征值（LAMBDA）和特征向量（V）"
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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": "ce8e4f8c-19a0-4392-b1ef-e794dcc5f187",
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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. , 0.5])"
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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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"LAMBDA"
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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": "13196da8-54c0-44bb-ac29-6cf7c6fec408",
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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([[ 0.70710678, 0.70710678],\n",
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" [-0.70710678, 0.70710678]])"
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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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"V"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c6bb1a72-57de-4ae9-a2e2-599db3122538",
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"metadata": {},
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"source": [
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"## 构建矩阵B"
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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": "cb92d4fe-8443-473f-a61c-378630468024",
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"metadata": {},
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"outputs": [],
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"source": [
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"B = V @ np.diag(np.sqrt(LAMBDA)) @ np.linalg.inv(V) # 根据特征值和特征向量构建矩阵B"
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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": "da0f2576-d262-41cd-8324-208cf3df1c82",
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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([[ 1.06066017, -0.35355339],\n",
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" [-0.35355339, 1.06066017]])"
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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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"B"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d70a4893-0524-47aa-91ad-4ad9863a5c89",
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"metadata": {},
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"source": [
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"## 重构矩阵A并打印"
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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": "d07c6472-787a-44c2-9d20-99e4d070826a",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[ 1.25 -0.75]\n",
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" [-0.75 1.25]]\n"
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]
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}
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],
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"source": [
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"A_reproduced = B @ B.T # 通过矩阵B的转置乘积重构矩阵A\n",
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"print(A_reproduced) # 输出重构后的矩阵A"
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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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}
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263
Book4_Ch14_Python_Codes/Bk4_Ch14_02.ipynb
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263
Book4_Ch14_Python_Codes/Bk4_Ch14_02.ipynb
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File diff suppressed because one or more lines are too long
54
Book4_Ch14_Python_Codes/Streamlit_Bk4_Ch14_02.py
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54
Book4_Ch14_Python_Codes/Streamlit_Bk4_Ch14_02.py
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###############
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# Authored by Weisheng Jiang
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# Book 4 | From Basic Arithmetic to Machine Learning
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# Published and copyrighted by Tsinghua University Press
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# Beijing, China, 2025
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###############
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import numpy as np # 导入NumPy库，用于数值计算
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import streamlit as st # 导入Streamlit库，用于创建交互式Web应用
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import time # 导入time模块，用于添加延时效果
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# 定义状态转移矩阵 A
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A = np.matrix([[0.7, 0.2], # 第一行表示从鸡到鸡的概率为0.7，从鸡到兔子的概率为0.3
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[0.3, 0.8]]) # 第二行表示从兔子到鸡的概率为0.2，从兔子到兔子的概率为0.8
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# 在侧边栏中创建交互组件
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with st.sidebar:
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# 创建滑块，用于用户设置初始状态中鸡的比例
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pi_0_chicken = st.slider('Ratio of chicken:', # 滑块标题
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0.0, 1.0, step=0.1) # 范围从0到1，每次步进0.1
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pi_0_rabbit = 1 - pi_0_chicken # 计算兔子的比例，使鸡和兔子的比例和为1
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st.write('Ratio of rabbit: ' + str(round(pi_0_rabbit, 1))) # 显示兔子的比例，保留1位小数
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# 创建滑块，用于用户设置模拟的天数
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num_iterations = st.slider('Number of nights:', # 滑块标题
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20, 100, step=5) # 范围从20到100，每次步进5
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# 创建进度条和状态文本，用于显示迭代进度
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progress_bar = st.sidebar.progress(0) # 初始化进度条为0%
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status_text = st.sidebar.empty() # 创建一个空白的文本区域，用于显示进度百分比
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# 初始化状态向量，将用户输入的初始比例存入数组
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last_rows = np.array([[pi_0_chicken, pi_0_rabbit]]) # 初始状态为一个行向量，包含鸡和兔子的比例
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# 创建一个折线图，用于实时展示状态变化
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chart = st.line_chart(last_rows) # 初始化折线图，并将初始状态绘制到图上
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# 开始迭代模拟状态转移
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for i in range(1, num_iterations): # 循环从第1天到用户设置的总天数
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last_status = last_rows[-1, :] # 获取当前的状态向量（最后一行）
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new_rows = last_status @ A.T # 使用矩阵乘法计算下一个状态，转移矩阵取转置
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percent = (i + 1) * 100 / num_iterations # 计算当前完成的百分比
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# 更新进度条和状态文本
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status_text.text("%i%% Complete" % percent) # 显示当前完成的百分比
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chart.add_rows(new_rows) # 将新状态添加到折线图
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progress_bar.progress(i) # 更新进度条的值
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last_rows = new_rows # 更新最后的状态向量为当前计算的状态
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time.sleep(0.1) # 延时0.1秒，以便观察每次状态更新
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# 清空进度条，表示模拟完成
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progress_bar.empty() # 移除进度条
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119
Book4_Ch14_Python_Codes/Streamlit_Bk4_Ch14_03.py
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119
Book4_Ch14_Python_Codes/Streamlit_Bk4_Ch14_03.py
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###############
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# Authored by Weisheng Jiang
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# Book 4 | From Basic Arithmetic to Machine Learning
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# Published and copyrighted by Tsinghua University Press
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# Beijing, China, 2025
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###############
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import plotly.graph_objects as go # 导入 Plotly 的图形对象模块，用于创建复杂的图形
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import streamlit as st # 导入 Streamlit 库，用于创建交互式 Web 应用
|
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import numpy as np # 导入 NumPy，用于数值计算
|
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import plotly.express as px # 导入 Plotly Express，用于快速绘制图表
|
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import pandas as pd # 导入 Pandas，用于数据处理
|
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import sympy # 导入 SymPy，用于符号运算和公式化表达
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# 定义函数 bmatrix，将 NumPy 数组转换为 LaTeX 格式的矩阵表示
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def bmatrix(a):
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"""返回一个 LaTeX 矩阵表示"""
|
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if len(a.shape) > 2: # 检查输入是否为二维数组
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raise ValueError('bmatrix 函数最多显示二维矩阵') # 如果不是二维，抛出异常
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lines = str(a).replace('[', '').replace(']', '').splitlines() # 将数组转换为字符串并去掉方括号
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rv = [r'\begin{bmatrix}'] # 开始 LaTeX 矩阵的格式
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rv += [' ' + ' & '.join(l.split()) + r'\\' for l in lines] # 逐行添加 LaTeX 矩阵行
|
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rv += [r'\end{bmatrix}'] # 结束 LaTeX 矩阵的格式
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return '\n'.join(rv) # 返回拼接后的 LaTeX 字符串
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|
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# 创建 Streamlit 侧边栏，用于调整矩阵 A 的参数
|
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with st.sidebar:
|
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# 显示矩阵 A 的 LaTeX 表示
|
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st.latex(r'''
|
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A = \begin{bmatrix}
|
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a & b\\
|
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b & c
|
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\end{bmatrix}''')
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||||
|
||||
# 创建滑块，允许用户调整矩阵 A 的元素 a, b, c 的值
|
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a = st.slider('a', -2.0, 2.0, step=0.05, value=1.0) # 滑块用于调整 a 的值
|
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b = st.slider('b', -2.0, 2.0, step=0.05, value=0.0) # 滑块用于调整 b 的值
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c = st.slider('c', -2.0, 2.0, step=0.05, value=1.0) # 滑块用于调整 c 的值
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#%% 创建一个单位圆的点集
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theta_array = np.linspace(0, 2 * np.pi, 36) # 在 [0, 2π] 区间生成 36 个点，表示角度
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X = np.column_stack((np.cos(theta_array), # 用 cos 和 sin 创建单位圆上的点
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np.sin(theta_array)))
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# 创建矩阵 A
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A = np.array([[a, b], # 矩阵 A 的第一行
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[b, c]]) # 矩阵 A 的第二行
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# 显示单位圆的方程和线性变换后的方程
|
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st.latex(r'''z^Tz = 1''') # 显示单位圆的方程
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||||
st.latex(r'''x = Az''') # 显示线性变换的方程
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# 显示矩阵 A 的 LaTeX 表示
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st.latex('A =' + bmatrix(A))
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# 对单位圆的点集进行线性变换
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X_ = X @ A # 对单位圆上的点集 X 应用线性变换矩阵 A
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#%% 使用符号运算求解椭圆的方程
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x1, x2 = sympy.symbols('x1 x2') # 定义符号变量 x1 和 x2
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y1, y2 = sympy.symbols('y1 y2') # 定义符号变量 y1 和 y2
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x = np.array([[x1, x2]]).T # 定义符号向量 x
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y = np.array([[y1, y2]]).T # 定义符号向量 y
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||||
# 计算 Q 矩阵
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Q = np.linalg.inv(A @ A.T) # Q = (AA^T)^(-1)
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D, V = np.linalg.eig(Q) # 计算 Q 的特征值和特征向量
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D = np.diag(D) # 将特征值转化为对角矩阵
|
||||
|
||||
# 显示 Q 矩阵的分解
|
||||
st.latex(r'Q = \left( AA^T\right)^{-1} = ' + bmatrix(np.round(Q, 3))) # 显示 Q 矩阵
|
||||
st.latex(r'''Q = V \Lambda V^{T}''') # 显示特征分解公式
|
||||
st.latex(bmatrix(np.around(Q, decimals=3)) + '=' +
|
||||
bmatrix(np.around(V, decimals=3)) + '@' +
|
||||
bmatrix(np.around(D, decimals=3)) + '@' +
|
||||
bmatrix(np.around(V.T, decimals=3))) # 显示分解过程
|
||||
|
||||
# 定义单位圆和变换后的椭圆方程
|
||||
f_x = x.T @ np.round(Q, 3) @ x # 单位圆在 Q 矩阵下的方程
|
||||
f_y = y.T @ np.round(D, 3) @ y # 椭圆在对角矩阵 D 下的方程
|
||||
|
||||
# 显示椭圆方程
|
||||
from sympy import *
|
||||
st.write('The formula of the ellipse:') # 显示椭圆方程的标题
|
||||
st.latex(latex(simplify(f_x[0][0])) + ' = 1') # 显示椭圆方程
|
||||
st.write('The formula of the transformed ellipse:') # 显示变换后椭圆方程的标题
|
||||
st.latex(latex(simplify(f_y[0][0])) + ' = 1') # 显示变换后的椭圆方程
|
||||
|
||||
#%% 添加颜色信息到变换后的点集
|
||||
color_array = np.linspace(0, 1, len(X)) # 为每个点生成一个颜色值
|
||||
X_c = np.column_stack((X_, color_array)) # 将颜色信息添加到点集中
|
||||
df = pd.DataFrame(X_c, columns=['x1', 'x2', 'color']) # 将点集转换为 DataFrame 格式
|
||||
|
||||
#%% 绘制散点图
|
||||
fig = px.scatter(df, # 使用 Pandas 数据框作为数据源
|
||||
x="x1", # 横轴为 x1
|
||||
y="x2", # 纵轴为 x2
|
||||
color='color', # 根据颜色值为点上色
|
||||
color_continuous_scale=px.colors.sequential.Rainbow) # 使用彩虹色带
|
||||
|
||||
# 设置图形布局
|
||||
fig.update_layout(
|
||||
autosize=False, # 禁用自动调整尺寸
|
||||
width=500, # 图表宽度为 500 像素
|
||||
height=500) # 图表高度为 500 像素
|
||||
|
||||
# 添加横轴和纵轴的参考线
|
||||
fig.add_hline(y=0, line_color='black') # 添加黑色的水平参考线
|
||||
fig.add_vline(x=0, line_color='black') # 添加黑色的垂直参考线
|
||||
fig.update_layout(coloraxis_showscale=False) # 隐藏颜色条
|
||||
fig.update_xaxes(range=[-3, 3]) # 设置 x 轴范围
|
||||
fig.update_yaxes(range=[-3, 3]) # 设置 y 轴范围
|
||||
|
||||
# 在 Streamlit 页面上显示图表
|
||||
st.plotly_chart(fig)
|
||||
|
||||
|
||||
Reference in New Issue
Block a user