{
"cells": [
{
"cell_type": "code",
"execution_count": 134,
"metadata": {},
"outputs": [],
"source": [
"#全部行都能输出\n",
"from IPython.core.interactiveshell import InteractiveShell\n",
"InteractiveShell.ast_node_interactivity = \"all\""
]
},
{
"cell_type": "code",
"execution_count": 135,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# numpy矩阵模块matlib"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"NumPy 包包含一个 Matrix 库 numpy.matlib 。此模块的函数返回矩阵而不是返回 ndarray对象。 \n",
"矩阵是一个二维数组"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {},
"outputs": [],
"source": [
"import numpy.matlib"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.matlib.empty(shape) \n",
"函数返回一个新的矩阵，且不初始化元素。里面的初始化数据无意义"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1., 1., 1., 1., 1.]])"
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"m = np.matlib.empty(5)\n",
"m"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"numpy.matrix"
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/plain": [
"matrix([[1., 2.],\n",
" [3., 4.]])"
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"m = np.matlib.empty((2,2))\n",
"type(m)\n",
"m"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
"ename": "ValueError",
"evalue": "shape too large to be a matrix.",
"output_type": "error",
"traceback": [
"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[1;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[1;32m<ipython-input-67-35e333e4fda2>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mm\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmatlib\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mempty\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m4\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;31m# 三维的数组不会是矩阵\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 2\u001b[0m \u001b[0mtype\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mm\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 3\u001b[0m \u001b[0mm\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;32mD:\\Anaconda3\\lib\\site-packages\\numpy\\matlib.py\u001b[0m in \u001b[0;36mempty\u001b[1;34m(shape, dtype, order)\u001b[0m\n\u001b[0;32m 47\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 48\u001b[0m \"\"\"\n\u001b[1;32m---> 49\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mndarray\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmatrix\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mshape\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdtype\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0morder\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0morder\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 50\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 51\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mones\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mshape\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdtype\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;32mNone\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0morder\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'C'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;32mD:\\Anaconda3\\lib\\site-packages\\numpy\\matrixlib\\defmatrix.py\u001b[0m in \u001b[0;36m__array_finalize__\u001b[1;34m(self, obj)\u001b[0m\n\u001b[0;32m 180\u001b[0m \u001b[1;32mreturn\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 181\u001b[0m \u001b[1;32melif\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mndim\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m2\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 182\u001b[1;33m \u001b[1;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"shape too large to be a matrix.\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 183\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 184\u001b[0m \u001b[0mnewshape\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;31mValueError\u001b[0m: shape too large to be a matrix."
]
}
],
"source": [
"m = np.matlib.empty((2,2,4)) # 三维的数组不会是矩阵\n",
"type(m)\n",
"m"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.matlib.zeros(shape) \n",
"返回一个全0的矩阵"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[0., 0., 0., 0., 0.]])"
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.zeros(5)"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[0., 0., 0., 0.],\n",
" [0., 0., 0., 0.],\n",
" [0., 0., 0., 0.]])"
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.zeros((3, 4))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.matlib.ones(shape) \n",
"返回一个全1的矩阵"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1., 1., 1., 1., 1.],\n",
" [1., 1., 1., 1., 1.],\n",
" [1., 1., 1., 1., 1.]])"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.ones((3, 5))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.matlib.eye(行， 列=行， k偏移) \n",
"这个函数返回一个矩阵，对角线元素为 1，其他位置为零。"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1., 0., 0.],\n",
" [0., 1., 0.],\n",
" [0., 0., 1.]])"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.eye(3) # 列数默认等于行数"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1., 0., 0., 0.],\n",
" [0., 1., 0., 0.],\n",
" [0., 0., 1., 0.]])"
]
},
"execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.eye(3, 4)"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[0., 1., 0., 0.],\n",
" [0., 0., 1., 0.],\n",
" [0., 0., 0., 1.]])"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.eye(3, 4, 1) # k=1向上偏移1"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[0., 0., 0., 0.],\n",
" [1., 0., 0., 0.],\n",
" [0., 1., 0., 0.]])"
]
},
"execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.eye(3, 4, -1) # k=1向下偏移1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.matlib.identity(阶数) \n",
"函数返回给定大小的单位矩阵。单位矩阵是主对角线元素都为 1 的**方阵**。"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1., 0., 0.],\n",
" [0., 1., 0.],\n",
" [0., 0., 1.]])"
]
},
"execution_count": 75,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.identity(3)"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[1., 0., 0., 0., 0.],\n",
" [0., 1., 0., 0., 0.],\n",
" [0., 0., 1., 0., 0.],\n",
" [0., 0., 0., 1., 0.],\n",
" [0., 0., 0., 0., 1.]])"
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.identity(5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.matlib.rand(shape) \n",
"这个函数以[0, 1)的数组成的二维数组"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[0.70354477, 0.10355941, 0.10140223, 0.91209582]])"
]
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.rand(4)"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[0.71320341, 0.17103505, 0.81294741, 0.00272406, 0.14810309],\n",
" [0.57606954, 0.61993777, 0.98707075, 0.56956267, 0.99584741],\n",
" [0.83406569, 0.88845715, 0.50835887, 0.59140323, 0.20945002]])"
]
},
"execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matlib.rand((3, 5))"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {},
"outputs": [
{
"ename": "ValueError",
"evalue": "shape too large to be a matrix.",
"output_type": "error",
"traceback": [
"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[1;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[1;32m<ipython-input-79-78c430c99182>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmatlib\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m3\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m4\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m5\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;31m# 不能超出二维\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[1;32mD:\\Anaconda3\\lib\\site-packages\\numpy\\matlib.py\u001b[0m in \u001b[0;36mrand\u001b[1;34m(*args)\u001b[0m\n\u001b[0;32m 261\u001b[0m \u001b[1;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mtuple\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 262\u001b[0m \u001b[0margs\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0margs\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 263\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0masmatrix\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 264\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 265\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mrandn\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;32mD:\\Anaconda3\\lib\\site-packages\\numpy\\matrixlib\\defmatrix.py\u001b[0m in \u001b[0;36masmatrix\u001b[1;34m(data, dtype)\u001b[0m\n\u001b[0;32m 69\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 70\u001b[0m \"\"\"\n\u001b[1;32m---> 71\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mmatrix\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdata\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdtype\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mdtype\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcopy\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;32mFalse\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 72\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 73\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;32mD:\\Anaconda3\\lib\\site-packages\\numpy\\matrixlib\\defmatrix.py\u001b[0m in \u001b[0;36m__new__\u001b[1;34m(subtype, data, dtype, copy)\u001b[0m\n\u001b[0;32m 135\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 136\u001b[0m \u001b[0mintype\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mN\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdtype\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdtype\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 137\u001b[1;33m \u001b[0mnew\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdata\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mview\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0msubtype\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 138\u001b[0m \u001b[1;32mif\u001b[0m \u001b[0mintype\u001b[0m \u001b[1;33m!=\u001b[0m \u001b[0mdata\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdtype\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 139\u001b[0m \u001b[1;32mreturn\u001b[0m \u001b[0mnew\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mastype\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mintype\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;32mD:\\Anaconda3\\lib\\site-packages\\numpy\\matrixlib\\defmatrix.py\u001b[0m in \u001b[0;36m__array_finalize__\u001b[1;34m(self, obj)\u001b[0m\n\u001b[0;32m 180\u001b[0m \u001b[1;32mreturn\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 181\u001b[0m \u001b[1;32melif\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mndim\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m2\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 182\u001b[1;33m \u001b[1;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"shape too large to be a matrix.\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 183\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 184\u001b[0m \u001b[0mnewshape\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;31mValueError\u001b[0m: shape too large to be a matrix."
]
}
],
"source": [
"np.matlib.rand((3, 4, 5)) # 不能超出二维"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# numpy线性代数模块linalg"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"NumPy 包包含 numpy.linalg 模块，提供线性代数所需的所有功能。\n",
"- numpy.linalg.det() 计算输入矩阵的行列式\n",
"- numpy.linalg.solve() 求解矩阵形式的线性方程的解\n",
"- numpy.linalg.inv() 计算矩阵的逆"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.matmul(a, b)\n",
"函数返回两个数组的矩阵乘积"
]
},
{
"cell_type": "code",
"execution_count": 181,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1, 2],\n",
" [3, 4]])"
]
},
"execution_count": 181,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/plain": [
"array([[11, 12],\n",
" [13, 14]])"
]
},
"execution_count": 181,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.array([[1,2],[3,4]])\n",
"b = np.array([[11,12],[13,14]])\n",
"a\n",
"b"
]
},
{
"cell_type": "code",
"execution_count": 182,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[37, 40],\n",
" [85, 92]])"
]
},
"execution_count": 182,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matmul(a, b)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.linalg.det(a)\n",
"求矩阵的行列式 \n",
"只有方阵才有对应的行列式"
]
},
{
"cell_type": "code",
"execution_count": 183,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1, 2],\n",
" [3, 4]])"
]
},
"execution_count": 183,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.array([[1, 2], [3, 4]])\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": 184,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-2.0000000000000004"
]
},
"execution_count": 184,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.linalg.det(a) "
]
},
{
"cell_type": "code",
"execution_count": 185,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-2"
]
},
"execution_count": 185,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1*4-2*3"
]
},
{
"cell_type": "code",
"execution_count": 186,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1, 2, 3],\n",
" [1, 3, 3],\n",
" [1, 0, 3]])"
]
},
"execution_count": 186,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.array([[1, 2, 3], [1, 3, 3], [1, 0, 3]])\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": 187,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.0"
]
},
"execution_count": 187,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.linalg.det(a) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**行列式等于0，说明存在完全的线性相关性**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.linalg.solve()\n",
"该函数给出了矩阵形式的线性方程的解。 \n",
"例如，考虑以下线性方程： \n",
"`x + y + z = 6` \n",
"`2y + 5z = -4` \n",
"`2x + 5y - z = 27` \n",
"\n",
"可以使用矩阵表示为：\n",
"\n",
"$\n",
"\\left[\n",
" \\begin{matrix}\n",
" 1 & 1 & 1 \\\\\n",
" 0 & 2 & 5 \\\\\n",
" 2 & 5 & -1\n",
" \\end{matrix}\n",
" \\right]\n",
" $\n",
".\n",
"$\n",
"\\left[\n",
" \\begin{matrix}\n",
" x \\\\\n",
" y \\\\\n",
" z \n",
" \\end{matrix}\n",
" \\right]\n",
"$\n",
"=\n",
"$\n",
"\\left[\n",
" \\begin{matrix}\n",
" 6 \\\\\n",
" -4 \\\\\n",
" 27 \n",
" \\end{matrix}\n",
" \\right]\n",
" $ \n",
"方程变为：AX = B \n",
"A是一个矩阵， 即一个二维数组 \n",
"B是一个一维的数组\n"
]
},
{
"cell_type": "code",
"execution_count": 188,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 1, 1, 1],\n",
" [ 0, 2, 5],\n",
" [ 2, 5, -1]])"
]
},
"execution_count": 188,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = np.array([[1, 1, 1], [0, 2, 5], [2, 5, -1]])\n",
"A"
]
},
{
"cell_type": "code",
"execution_count": 189,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 6, -4, 27])"
]
},
"execution_count": 189,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B = np.array([6, -4, 27])\n",
"B"
]
},
{
"cell_type": "code",
"execution_count": 190,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 5., 3., -2.])"
]
},
"execution_count": 190,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.linalg.solve(A, B)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"则解出： x=5, y=3, z=-2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"可以利用矩阵乘法验证下"
]
},
{
"cell_type": "code",
"execution_count": 191,
"metadata": {},
"outputs": [],
"source": [
"a = np.linalg.solve(A, B) "
]
},
{
"cell_type": "code",
"execution_count": 192,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 6., -4., 27.])"
]
},
"execution_count": 192,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.matmul(A, a)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## np.linalg.inv(a) \n",
"返回矩阵的逆矩阵"
]
},
{
"cell_type": "code",
"execution_count": 193,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1, 2],\n",
" [3, 4]])"
]
},
"execution_count": 193,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.array([[1,2],[3,4]])\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": 194,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[-2. , 1. ],\n",
" [ 1.5, -0.5]])"
]
},
"execution_count": 194,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.linalg.inv(a)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
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