{
"cells": [
{
"attachments": {},
"cell_type": "markdown",
"id": "8d3c680b",
"metadata": {},
"source": [
"# 线性代数模块\n",
"\n",
"scipy.linalg是SciPy中负责线性代数计算的模块。NumPy也有numpy.linalg模块，不过建议使用scipy.linalg，原因有二：\n",
"- scipy.linalg包含numpy.linalg中的所有函数，同时还包含了很多numpy.linalg中没有的函数；\n",
"- scipy.linalg默认使用线性代数库BLAS/LAPACK等对运算进行加速，而在numpy.linalg中，这些加速是需要自己配置，不是默认设置:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "aa0ef52c",
"metadata": {},
"outputs": [],
"source": [
"import numpy.linalg"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "1cba1a1a",
"metadata": {},
"outputs": [],
"source": [
"import scipy.linalg"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "b65ef6f6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"34"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"len(dir(numpy.linalg))"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "4a139a95",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"156"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"len(dir(scipy.linalg))"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "c920896b",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "571cb2e1",
"metadata": {},
"outputs": [],
"source": [
"from scipy import linalg"
]
},
{
"cell_type": "markdown",
"id": "79b71214",
"metadata": {},
"source": [
"## 基本的矩阵操作"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "969794d4",
"metadata": {},
"source": [
"在NumPy中，矩阵有两种表示方法：矩阵类型和二维数组类型。\n",
"\n",
"矩阵类型可以用`np.mat()`或者`np.matrix()`创建："
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "fa1f33c1",
"metadata": {},
"outputs": [],
"source": [
"A = np.mat(\"[1, 2; 3, 4]\")"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "094e37a6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1, 2],\n",
" [3, 4]])"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "d221a93a",
"metadata": {},
"outputs": [],
"source": [
"A = np.matrix(\"[1, 2; 3, 4]\")"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "ffb1da83",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1, 2],\n",
" [3, 4]])"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "2e135a8f",
"metadata": {},
"source": [
"转置矩阵为："
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "5544773c",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1, 3],\n",
" [2, 4]])"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.T"
]
},
{
"cell_type": "markdown",
"id": "755aa7fa",
"metadata": {},
"source": [
"矩阵的逆矩阵："
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "d895134e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[-2. , 1. ],\n",
" [ 1.5, -0.5]])"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.I"
]
},
{
"cell_type": "markdown",
"id": "e19f8bcc",
"metadata": {},
"source": [
"矩阵乘法："
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "4a6607e0",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1.00000000e+00, 0.00000000e+00],\n",
" [1.11022302e-16, 1.00000000e+00]])"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.I * A"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "23bd32db",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1.0000000e+00, 0.0000000e+00],\n",
" [8.8817842e-16, 1.0000000e+00]])"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A * A.I"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "3509019e",
"metadata": {},
"source": [
"数组类型也可以完成类似的操作："
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "89a53b06",
"metadata": {},
"outputs": [],
"source": [
"A = np.array([[1, 2], [3, 4]])"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "eab9d269",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1, 2],\n",
" [3, 4]])"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A"
]
},
{
"cell_type": "markdown",
"id": "a504404a",
"metadata": {},
"source": [
"转置："
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "9b50a2c0",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1, 3],\n",
" [2, 4]])"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.T"
]
},
{
"cell_type": "markdown",
"id": "411b9f36",
"metadata": {},
"source": [
"其逆矩阵可以用`linalg.inv()`函数计算："
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "82bc4036",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1.0000000e+00, 0.0000000e+00],\n",
" [8.8817842e-16, 1.0000000e+00]])"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.dot(linalg.inv(A))"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "9a52cc39",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1.0000000e+00, 0.0000000e+00],\n",
" [8.8817842e-16, 1.0000000e+00]])"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A @ linalg.inv(A)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "e871f638",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1.00000000e+00, 0.00000000e+00],\n",
" [1.11022302e-16, 1.00000000e+00]])"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.inv(A) @ A"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "250c4d3f",
"metadata": {},
"source": [
"`scipy.linalg`也可以作用在矩阵类型上，返回的还是一个数组："
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "0b0ea345",
"metadata": {},
"outputs": [],
"source": [
"A = np.matrix([[1, 2], [3, 4]])"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "fcb3602e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[-2. , 1. ],\n",
" [ 1.5, -0.5]])"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.inv(A)"
]
},
{
"cell_type": "markdown",
"id": "f9397b1c",
"metadata": {},
"source": [
"矩阵类型的使用比较少，通常都用数组类型替代矩阵类型进行计算。"
]
},
{
"cell_type": "markdown",
"id": "e297badc",
"metadata": {},
"source": [
"## 线性方程组的求解"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "a24b6a4d",
"metadata": {},
"source": [
"考虑下列方程组：\n",
"```\n",
" x + 3y + 5z = 10\n",
"2x + 5y - z = 8\n",
"2x + 3y + 8z = 3\n",
"```\n",
"\n",
"矩阵形式为：\n",
"$$\n",
"\\left[\n",
"\\begin{array}{1}\n",
"1 & 3 & 5 \\\\\n",
"2 & 5 & -1 \\\\\n",
"2 & 3 & 8 \\\\\n",
"\\end{array}\n",
"\\right]\n",
"\\left[\n",
"\\begin{array}{1}\n",
"x \\\\\n",
"y \\\\\n",
"z \\\\\n",
"\\end{array}\n",
"\\right]\n",
"=\n",
"\\left[\n",
"\\begin{array}{1}\n",
"10 \\\\\n",
"8 \\\\\n",
"3 \\\\\n",
"\\end{array}\n",
"\\right]\n",
"$$\n",
"\n",
"其解为：\n",
"$$\n",
"\\left[\n",
"\\begin{array}{1}\n",
"x \\\\\n",
"y \\\\\n",
"z \\\\\n",
"\\end{array}\n",
"\\right]\n",
"=\n",
"\\left[\n",
"\\begin{array}{1}\n",
"1 & 3 & 5 \\\\\n",
"2 & 5 & -1 \\\\\n",
"2 & 3 & 8 \\\\\n",
"\\end{array}\n",
"\\right]^{-1}\n",
"\\left[\n",
"\\begin{array}{1}\n",
"10 \\\\\n",
"8 \\\\\n",
"3 \\\\\n",
"\\end{array}\n",
"\\right]\n",
"$$\n",
"\n",
"故："
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "a631aeb7",
"metadata": {},
"outputs": [],
"source": [
"A = np.array([[1, 3, 5],\n",
" [2, 5, -1],\n",
" [2, 3, 8]])"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "3017b816",
"metadata": {},
"outputs": [],
"source": [
"b = np.array([10, 8, 3])"
]
},
{
"cell_type": "markdown",
"id": "af0e6885",
"metadata": {},
"source": [
"解："
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "64da5739",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([-8.83870968, 5.25806452, 0.61290323])"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.inv(A) @ b"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "3ff7b6d3",
"metadata": {},
"source": [
"也可以直接使用`linalg.solve()`函数，直接求解这个方程："
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "349caf78",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([-8.83870968, 5.25806452, 0.61290323])"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.solve(A, b)"
]
},
{
"cell_type": "markdown",
"id": "19f5a3b0",
"metadata": {},
"source": [
"## 行列式的计算："
]
},
{
"cell_type": "markdown",
"id": "00a11f84",
"metadata": {},
"source": [
"`linalg.det()`函数可以计算矩阵的行列式："
]
},
{
"cell_type": "code",
"execution_count": 27,
"id": "73f30cf7",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-31.000000000000007"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.det(A)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "a392795e",
"metadata": {},
"source": [
"行列式非零表示矩阵是可逆的。逆矩阵的行列式与矩阵的行列式满足相乘等于1的关系："
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "e2989093",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-0.03225806451612903"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.det(linalg.inv(A))"
]
},
{
"cell_type": "code",
"execution_count": 29,
"id": "7762fa35",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.0000000000000002"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.det(linalg.inv(A)) * linalg.det(A)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "e46a7a69",
"metadata": {},
"source": [
"## 范数的计算"
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "9b9bf730",
"metadata": {},
"outputs": [],
"source": [
"A = np.array([[1, 2], [3, 4]])"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "db9beb30",
"metadata": {},
"source": [
"默认情况下，`linalg.norm()`函数计算的是矩阵的Frobenius范数，该范数计算的是矩阵所有元素平方和的平方根："
]
},
{
"cell_type": "code",
"execution_count": 31,
"id": "e09c0730",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5.477225575051661"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(A)"
]
},
{
"cell_type": "code",
"execution_count": 32,
"id": "658a1746",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5.477225575051661"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(A, \"fro\")"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "a0d0a4a5",
"metadata": {},
"source": [
"可以通过`linalg.norm()`函数中的第二个参数，来修改矩阵范数的计算方法。例如，矩阵的1范数，计算的是矩阵中每列元素的和的最大值："
]
},
{
"cell_type": "code",
"execution_count": 33,
"id": "affe6ca1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6.0"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(A, 1)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "0d8fca3b",
"metadata": {},
"source": [
"矩阵的-1范数，计算的是矩阵中每列元素的和的最小值："
]
},
{
"cell_type": "code",
"execution_count": 34,
"id": "8eed8d80",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4.0"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(A, -1)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "91357c89",
"metadata": {},
"source": [
"矩阵的2范数，计算的是矩阵的最大奇异值："
]
},
{
"cell_type": "code",
"execution_count": 35,
"id": "d0fa7c53",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5.464985704219043"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(A, 2)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "b3421820",
"metadata": {},
"source": [
"矩阵的-2范数，计算的是矩阵的最小奇异值："
]
},
{
"cell_type": "code",
"execution_count": 36,
"id": "14389d34",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.36596619062625746"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(A, -2)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "da8e92f7",
"metadata": {},
"source": [
"矩阵的正无穷范数，计算的是矩阵中每行元素的和的最大值："
]
},
{
"cell_type": "code",
"execution_count": 37,
"id": "18293ca2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"7.0"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(A, np.inf)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "6c6ea89e",
"metadata": {},
"source": [
"矩阵的负无穷范数，计算的是矩阵中每行元素的和的最小值："
]
},
{
"cell_type": "code",
"execution_count": 38,
"id": "055a13b9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3.0"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(A, -np.inf)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "b99a6ae5",
"metadata": {},
"source": [
"`linalg.norm()`函数也可以用来计算向量的范数（也称模）："
]
},
{
"cell_type": "code",
"execution_count": 39,
"id": "5b01a21d",
"metadata": {},
"outputs": [],
"source": [
"b = np.array([10, 8, 3])"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "a3ffeddd",
"metadata": {},
"source": [
"对于向量来说，linalg.norm()函数默认是向量的2范数，即计算向量元素的平方和的平方根："
]
},
{
"cell_type": "code",
"execution_count": 40,
"id": "40c8835e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"13.152946437965905"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(b)"
]
},
{
"cell_type": "code",
"execution_count": 41,
"id": "0bbce740",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"13.152946437965905"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linalg.norm(b, 2)"
]
},
{
"cell_type": "markdown",
"id": "3e57bf3d",
"metadata": {},
"source": [
"## 特征值分解"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "35edb52d",
"metadata": {},
"source": [
"对于方阵A，特征值λ和特征向量v满足：`Av=λv`。函数`linalg.eig()`可以用来求解特征值和特征向量："
]
},
{
"cell_type": "code",
"execution_count": 42,
"id": "498f619f",
"metadata": {},
"outputs": [],
"source": [
"A = np.array([[1, 2, 3],\n",
" [4, 5, 6],\n",
" [7, 8, 9]])"
]
},
{
"cell_type": "code",
"execution_count": 43,
"id": "4b470fd3",
"metadata": {},
"outputs": [],
"source": [
"l, v = linalg.eig(A)"
]
},
{
"cell_type": "code",
"execution_count": 44,
"id": "7695b549",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.allclose(A.dot(v), l * v)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "a5623725",
"metadata": {},
"source": [
"## 广义逆\n",
"\n",
"对于形状为m×n的矩阵A，可以用`linalg.pinv()`求解其广义逆。A的广义逆B满足：`A=ABA`，`B=ABA`："
]
},
{
"cell_type": "code",
"execution_count": 45,
"id": "3e65543f",
"metadata": {},
"outputs": [],
"source": [
"A = np.array([[1, 2, 3],\n",
" [4, 5, 6]])"
]
},
{
"cell_type": "code",
"execution_count": 46,
"id": "f7bc6b49",
"metadata": {},
"outputs": [],
"source": [
"B = linalg.pinv(A)"
]
},
{
"cell_type": "code",
"execution_count": 47,
"id": "ac4d5781",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.allclose(A, A @ B @ A)"
]
},
{
"cell_type": "code",
"execution_count": 48,
"id": "ea20c7e1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.allclose(B, B @ A @ B)"
]
},
{
"cell_type": "markdown",
"id": "7463da00",
"metadata": {},
"source": [
"## 奇异值分解"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "74047d9c",
"metadata": {},
"source": [
"对于形状为m×n的矩阵A，其奇异值分解满足$A=U\\SigmaV^H$，矩阵Σ形状为M×N，主对角线上的元素被称为奇异值。U、V分别为M阶、N阶正交单位矩阵。\n",
"\n",
"函数`linalg.svd()`可以对矩阵进行奇异值分解："
]
},
{
"cell_type": "code",
"execution_count": 49,
"id": "ffae72b3",
"metadata": {},
"outputs": [],
"source": [
"U, s, Vh = linalg.svd(A)"
]
},
{
"cell_type": "code",
"execution_count": 50,
"id": "be9437a5",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[-0.3863177 , -0.92236578],\n",
" [-0.92236578, 0.3863177 ]])"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U"
]
},
{
"cell_type": "code",
"execution_count": 51,
"id": "1af28a78",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([9.508032 , 0.77286964])"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"s"
]
},
{
"cell_type": "code",
"execution_count": 52,
"id": "b9174266",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[-0.42866713, -0.56630692, -0.7039467 ],\n",
" [ 0.80596391, 0.11238241, -0.58119908],\n",
" [ 0.40824829, -0.81649658, 0.40824829]])"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Vh"
]
},
{
"cell_type": "markdown",
"id": "cd3bdbca",
"metadata": {},
"source": [
"从奇异值恢复矩阵$\\Sigma$："
]
},
{
"cell_type": "code",
"execution_count": 53,
"id": "26907656",
"metadata": {},
"outputs": [],
"source": [
"S = linalg.diagsvd(s, 2, 3)"
]
},
{
"cell_type": "code",
"execution_count": 54,
"id": "c2878f35",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.allclose(A, U @ S @ Vh)"
]
},
{
"cell_type": "markdown",
"id": "cd63330a",
"metadata": {},
"source": [
"正交矩阵："
]
},
{
"cell_type": "code",
"execution_count": 55,
"id": "c0ae03d2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 1.00000000e+00, -8.49433239e-18],\n",
" [-8.49433239e-18, 1.00000000e+00]])"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U @ U.T"
]
},
{
"cell_type": "code",
"execution_count": 56,
"id": "dec8f9d2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 1.00000000e+00, 4.13108370e-17, 2.21870637e-17],\n",
" [ 4.13108370e-17, 1.00000000e+00, -1.77008767e-16],\n",
" [ 2.21870637e-17, -1.77008767e-16, 1.00000000e+00]])"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Vh @ Vh.T"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f1d161a1",
"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": "3.9.10"
}
},
"nbformat": 4,
"nbformat_minor": 5
}