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Commit fee054d

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math
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‎math/求导法则.ipynb‎

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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 行向量对元素求导\n",
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"\n",
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"设$y^T=[y_1 ... y_n]$是n维行向量,$x$是元素\n",
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"\n",
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"$\\frac{\\partial y^T}{\\partial x}=[ \\frac{\\partial y_1}{\\partial x} ... \\frac{\\partial y_n}{\\partial x} ] $\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 列向量对元素求导\n",
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"\n",
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"设$y=\\begin{bmatrix}y_1\\\\ ...\\\\ y_n\\end{bmatrix}$是m维列向量,$x$是元素\n",
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"\n",
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"则$\\frac{\\partial y}{\\partial x}=\\begin{bmatrix} \\frac{\\partial y_1}{\\partial x}\\\\ ... \\\\\n",
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"\\frac{\\partial y_m}{\\partial x} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 矩阵对元素求导\n",
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"\n",
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"设$Y=\\begin{bmatrix}y_1 & ... & y_{1n}\\\\...&...& ...\\\\ y_{m1} & ... & y_{mn}\\end{bmatrix}$是$m*n$维矩阵,$x$是元素\n",
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"\n",
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"则$\\frac{\\partial Y}{\\partial x}=\\begin{bmatrix} \\frac{\\partial y_{11}}{\\partial x} & ... &\\frac{\\partial y_{1n}}{\\partial x}\\\\...&...& ... \\\\\n",
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"\\frac{\\partial y_{m1}}{\\partial x}& ... & \\frac{\\partial y_{mn}}{\\partial x} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 元素对行向量求导\n",
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"\n",
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"设$y$是元素,$x^T=[x_1 ... x_q]$是q维行向量\n",
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"\n",
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"则$\\frac{\\partial y}{\\partial x^T}=[ \\frac{\\partial y}{\\partial x_1} ... \\frac{\\partial y}{\\partial x_q} ] $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 元素对列向量求导\n",
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"\n",
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"设$y$是元素,$x=\\begin{bmatrix}x_1\\\\ ...\\\\ x_n\\end{bmatrix}$是p维列向量\n",
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"\n",
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"则$\\frac{\\partial y}{\\partial x}=\\begin{bmatrix} \\frac{\\partial y}{\\partial x_1}\\\\ ... \\\\\n",
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"\\frac{\\partial y}{\\partial x_p} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 元素对矩阵求导\n",
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"\n",
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"设$y$是元素,$X=\\begin{bmatrix}x_1 & ... & x_{1q}\\\\...&...& ...\\\\ x_{p1} & ... & x_{pq}\\end{bmatrix}$是$p*q$维矩阵\n",
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"\n",
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"则$\\frac{\\partial y}{\\partial X}=\\begin{bmatrix} \\frac{\\partial y}{\\partial x_{11}} & ... &\\frac{\\partial y}{\\partial x_{1q}}\\\\...&...& ... \\\\\n",
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"\\frac{\\partial y}{\\partial x_{p1}}& ... & \\frac{\\partial y}{\\partial x_{pq}} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 行向量对列向量求导\n",
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"\n",
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"设$y^T=[y_1 ... y_n]$是n维行向量,$x=\\begin{bmatrix}x_1\\\\ ...\\\\ x_p\\end{bmatrix}$是p维列向量\n",
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"\n",
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"则$\\frac{\\partial y^T}{\\partial x}=\\begin{bmatrix} \\frac{\\partial y_{1}}{\\partial x_{1}} & ... &\\frac{\\partial y_{n}}{\\partial x_{1}}\\\\...&...& ... \\\\\n",
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"\\frac{\\partial y_1}{\\partial x_{p}}& ... & \\frac{\\partial y_n}{\\partial x_{p}} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 列向量对行向量求导\n",
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"\n",
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"设$y=\\begin{bmatrix}y_1\\\\ ...\\\\ y_m\\end{bmatrix}$是m维列向量,$x^T=[x_1 ... x_q]$是q维行向量\n",
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"\n",
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"则$\\frac{\\partial y}{\\partial x^T}=\\begin{bmatrix} \\frac{\\partial y_{1}}{\\partial x_{1}} & ... &\\frac{\\partial y_{1}}{\\partial x_{q}}\\\\...&...& ... \\\\\n",
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"\\frac{\\partial y_m}{\\partial x_{1}}& ... & \\frac{\\partial y_m}{\\partial x_{q}} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 行向量对行向量求导\n",
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"\n",
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"设$y^T=[y_1 ... y_n]$是n维行向量,$x^T=[x_1 ... x_q]$是q维行向量\n",
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"\n",
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"则$\\frac{\\partial y^T}{\\partial x^T}=[ \\frac{\\partial y^T}{\\partial x_1} ... \\frac{\\partial y^T}{\\partial x_q} ] $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 列向量对列向量求导\n",
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"\n",
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"设$y=\\begin{bmatrix}y_1\\\\ ...\\\\ y_m\\end{bmatrix}$是m维列向量,$x=\\begin{bmatrix}x_1\\\\ ...\\\\ x_p\\end{bmatrix}$是p维列向量\n",
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"\n",
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"则$\\frac{\\partial y}{\\partial x}=\\begin{bmatrix} \\frac{\\partial y_1}{\\partial x} \\\\...\\\\\n",
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"\\frac{\\partial y_m}{\\partial x} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 矩阵对行向量求导\n",
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"\n",
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"设$Y=\\begin{bmatrix}y_1 & ... & y_{1n}\\\\...&...& ...\\\\ y_{m1} & ... & y_{mn}\\end{bmatrix}$是$m*n$维矩阵,$x^T=[x_1 ... x_q]$是q维行向量\n",
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"\n",
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"则$\\frac{\\partial Y}{\\partial x^T}=[ \\frac{\\partial Y}{\\partial x_1} ... \\frac{\\partial Y}{\\partial x_q} ] $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 矩阵对列向量求导\n",
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"\n",
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"设$Y=\\begin{bmatrix}y_1 & ... & y_{1n}\\\\...&...& ...\\\\ y_{m1} & ... & y_{mn}\\end{bmatrix}$是$m*n$维矩阵,$x=\\begin{bmatrix}x_1\\\\ ...\\\\ x_n\\end{bmatrix}$是p维列向量\n",
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"\n",
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"则$\\frac{\\partial Y}{\\partial x}=\\begin{bmatrix} \\frac{\\partial y_{11}}{\\partial x} & ... &\\frac{\\partial y_{1n}}{\\partial x}\\\\...&...& ... \\\\\n",
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"\\frac{\\partial y_{m1}}{\\partial x}& ... & \\frac{\\partial y_{mn}}{\\partial x} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 行向量对矩阵求导\n",
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"\n",
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"设$y^T=[y_1 ... y_n]$是n维行向量,$X=\\begin{bmatrix}x_1 & ... & x_{1q}\\\\...&...& ...\\\\ x_{p1} & ... & x_{pq}\\end{bmatrix}$是$p*q$维矩阵\n",
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"\n",
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"则$\\frac{\\partial y^T}{\\partial X}=\\begin{bmatrix} \\frac{\\partial y^T}{\\partial x_{11}} & ... &\\frac{\\partial y^T}{\\partial x_{1q}}\\\\...&...& ... \\\\\n",
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"\\frac{\\partial y^T}{\\partial x_{p1}}& ... & \\frac{\\partial y^T}{\\partial x_{pq}} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 列向量对矩阵求导\n",
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"\n",
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"设$y=\\begin{bmatrix}y_1\\\\ ...\\\\ y_m\\end{bmatrix}$是m维列向量,$X=\\begin{bmatrix}x_1 & ... & x_{1q}\\\\...&...& ...\\\\ x_{p1} & ... & x_{pq}\\end{bmatrix}$是$p*q$维矩阵\n",
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"\n",
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"则$\\frac{\\partial y}{\\partial X}=\\begin{bmatrix} \\frac{\\partial y_1}{\\partial X}\\\\ ... \\\\\n",
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"\\frac{\\partial y_m}{\\partial X} \\end{bmatrix} $"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 矩阵对矩阵求导\n",
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"\n",
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"设$Y=\\begin{bmatrix}y_1 & ... & y_{1n}\\\\...&...& ...\\\\ y_{m1} & ... & y_{mn}\\end{bmatrix}=\\begin{bmatrix}y_1^T\\\\ ...\\\\ y_m^T\\end{bmatrix}$是$m*n$维矩阵,$X=\\begin{bmatrix}x_1 & ... & x_{1q}\\\\...&...& ...\\\\ x_{p1} & ... & x_{pq}\\end{bmatrix}=[x_1 ... x_q]$是$p*q$维矩阵\n",
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"\n",
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"则$\\frac{\\partial Y}{\\partial X}\n",
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"=[ \\frac{\\partial Y}{\\partial x_1} ... \\frac{\\partial Y}{\\partial x_q} ]\n",
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"=\\begin{bmatrix}\n",
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"\\frac{\\partial y_1^T}{\\partial X}\\\\\n",
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"... \\\\\n",
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"\\frac{\\partial y_m^T}{\\partial X} \n",
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"\\end{bmatrix} \n",
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"=\\begin{bmatrix}\n",
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"\\frac{\\partial y_{1}^T}{\\partial x_{1}} & ... &\\frac{\\partial y_{1}^T}{\\partial x_{q}}\\\\\n",
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"...&...& ... \\\\\n",
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"\\frac{\\partial y_m^T}{\\partial x_{1}}& ... & \\frac{\\partial y_m^T}{\\partial x_{q}} \\end{bmatrix} $"
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]
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}
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],
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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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": "3.5.2"
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"toc": {
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"colors": {
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"hover_highlight": "#DAA520",
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"navigate_num": "#000000",
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"navigate_text": "#333333",
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"running_highlight": "#FF0000",
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"selected_highlight": "#FFD700",
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"sidebar_border": "#EEEEEE",
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"moveMenuLeft": true,
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"nav_menu": {
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"height": "282px",
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"width": "252px"
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"navigate_menu": true,
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"number_sections": true,
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"sideBar": true,
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"threshold": 4,
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"toc_cell": false,
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"toc_section_display": "block",
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"toc_window_display": false,
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"nbformat": 4,
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"nbformat_minor": 2
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}

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