「統計解析のための線形代数」復習筆記2

偏微分

1個變量的函數的微分

公式:

  1. 函數 \(f(a)\) 關於變量 \(a\) 的微分,被定義爲: \(\lim\limits_{h \to 0} \frac{f(a+h)-f(a)}{h}\) , 寫作 \(\frac{df}{da}\), 具有下列性質:
    • \(f(a) = a^n\) 時, \(\frac{df}{da} = na^{n-1}\) 重要
    • \(\frac{d}{da}\left\{kf(a)+lg(a)\right\}=k\frac{df}{da}+l\frac{dg}{da}\) (\(k,l\) 是常數)
    • \(\frac{d}{da}\left\{f(a) \cdot g(a)\right\}=\frac{df}{da}g(a)+f{a}\frac{dg}{da}\)
    • \(\frac{d}{da}\left\{\frac{f(a)}{g(a)}\right\}=\frac{\frac{df}{da}g(a)-f(a)\frac{dg}{da}}{\left\{g(a)\right\}^2}\), 特別的有,\(\frac{d}{da}\left\{\frac{1}{g(a)}\right\}=-\frac{\frac{dg}{da}}{\left\{g(a)\right\}^2}\)
    • \(y=f(b), b=g(a)\) 時, \(\frac{dy}{da}=\frac{dy}{db}\frac{db}{da}\)
  2. 2次(2階)微分 【二階導數】:

    \(f(a)\) 關於常數 \(a\) 的微分 \(\frac{df}{da}\) 的二次微分表示爲: \(\frac{d^2f}{da^2}\)

多個變量的函數的微分

偏微分

包含了 \(n\) 個獨立變量 \(a_1, a_2,a_3,\cdots,a_i,\cdots,a_n\)的函數,即多變量函數 \(F(a_1, a_2,a_3,\cdots,a_i,\cdots,a_n)\) 關於 \(a_i (i=1,2,\cdots,n)\) 的偏微分 (partial differentiation) 的定義是,把 \(a_i\) 以外的獨立變量當做常數(定数),將函數 \(F\) 對變量 \(a_i\) 求微分,寫作: \(\frac{\partial F}{\partial a_i}\)

以下爲了便於說明,以三個變量爲例。

  1. 函數 \(F(a_1,a_2,a_3)=a_1+a_2+a_3=\sum\limits_{i=1}^3a_i\) 對於三個獨立變量分別求偏微分: \[\frac{\partial F}{\partial a_1}=1,\frac{\partial F}{\partial a_2}=1, \frac{\partial F}{\partial a_3}=1\]

  2. 函數 \(F(a_1,a_2,a_3)=a_1b_1+a_2b_2+a_3b_3=\sum\limits_{i=1}^3a_ib_i\) 對於三個獨立變量分別求偏微分: \[\frac{\partial F}{\partial a_1}=b_1,\frac{\partial F}{\partial a_2}=b_2, \frac{\partial F}{\partial a_3}=b_3\]

  3. 函數 \(F(a_1,a_2,a_3)=a_1^2+a_2^2+a_3^2=a_1\cdot a_1+a_2\cdot a_2+a_3\cdot a_3\\=\sum\limits_{i=1}^3a_i^2=\sum\limits_{i=1}^3a_i\cdot a_i \;對三個變量分別求偏微分:\)  \[\frac{\partial F}{\partial a_1}=2a_1,\frac{\partial F}{\partial a_2}=2a_2, \frac{\partial F}{\partial a_3}=2a_3\]
  4. 函數 \(F(a_1,a_2,a_3)=\lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2=a_1\cdot\lambda_1\cdot a_1+a_2\cdot\lambda_2\cdot a_2+a_3\cdot\lambda_3\cdot a_3\\=\sum\limits_{i=1}^3\lambda_ia_i^2=\sum\limits_{i=1}^3a_i\cdot\lambda_i\cdot a_i \; 對三個變量分別求偏微分:\) \[\frac{\partial F}{\partial a_1}=2\lambda_1a_1,\frac{\partial F}{\partial a_2}=2\lambda_2a_2, \frac{\partial F}{\partial a_3}=2\lambda_3a_3\]
  5. 函數 \(F(a_1,a_2,a_3)=(b_1-\lambda a_1)^2+(b_2-\lambda a_2)^2+(b_3-\lambda a_3)^2\\=\sum\limits_{i=1}^3(b_i-\lambda a_i)^2=\sum\limits_{i=1}^3(b_i-\lambda a_i)(b_i-\lambda a_i)\;對三個變量求偏微分:\) \[\frac{\partial F}{\partial a_1}=-2\lambda(b_1-\lambda a_1),\frac{\partial F}{\partial a_2}=-2\lambda(b_2-\lambda a_2), \frac{\partial F}{\partial a_3}=-2\lambda(b_3-\lambda a_3)\]
  6. 函數 \(F = a_{11}x_1y_1 + a_{12}x_1y_2 + a_{13}x_1y_3 \\ \;\;\;\;\;\;+a_{21}x_2y_1+a_{22}x_2y_2+a_{23}x_2y_3\\ \;\;\;\;\;\;+a_{31}x_3y_1+a_{32}x_3y_2+a_{33}x_3y_3\\ \;\;\;=\sum\limits_{i=1}^3\sum\limits_{i=1}^3a_{ij}x_iy_j\;對三個變量求偏微分:\)
    • \(F(x_1,x_2,x_3)\), 即視爲 \(x_1,x_2,x_3\) 的函數的時候:

    \[ \frac{\partial F}{\partial x_1}=a_{11}y_1+a_{12}y_2+a_{13}y_3=\sum_{j=1}^3a_{1j}y_j \\ \frac{\partial F}{\partial x_2}=a_{21}y_1+a_{22}y_2+a_{23}y_3=\sum_{j=1}^3a_{2j}y_j \\ \frac{\partial F}{\partial x_3}=a_{31}y_1+a_{32}y_2+a_{33}y_3=\sum_{j=1}^3a_{3j}y_j \\ 將上面三個式子總結一下就是: \\ \frac{\partial F}{\partial x_i}=\sum_{j=1}^3a_{ij}y_j (i=1,2,3) \]

    • \(F(y_1,y_2,y_3)\), 即視爲 \(y_1,y_2,y_3\) 的函數的時候:

    \[ \frac{\partial F}{\partial y_1}=a_{11}x_1+a_{21}x_2+a_{31}x_3=\sum_{i=1}^3a_{i1}x_i \\ \frac{\partial F}{\partial y_2}=a_{12}x_1+a_{22}x_2+a_{32}x_3=\sum_{i=1}^3a_{i2}x_i \\ \frac{\partial F}{\partial y_3}=a_{13}x_1+a_{32}x_2+a_{33}x_3=\sum_{i=1}^3a_{i3}x_i \\ 將上面三個式子總結一下就是: \\ \frac{\partial F}{\partial x_i}=\sum_{i=1}^3a_{ij}x_i (j=1,2,3) \]

  7. 函數 \(F(x_1,x_2,x_3)=a_{11}x_1x_1+a_{12}x_1x_2+a_{13}x_1x_3 \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+a_{12}x_2x_1+a_{12}x_2x_2+a_{23}x_2x_3\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+a_{13}x_3x_1+a_{23}x_3x_2+a_{33}x_3x_3\\ \;\;\;\;\;\;\;\;\;\;\;\;=a_{11}x_1^2+2a_{12}x_1x_2+2a_{13}x_1x_3+a_{22}x_2^2+2a_{23}x_2x_3+a_{33}x_3^2\\ \;\;\;\;\;\;\;\;\;\;\;\;=\sum\limits_{i=1}^3a_{ii}x_i^2+2\mathop{\sum\limits^3\sum\limits^3}\limits_{i<j}a_{ij}x_ix_j\\ \;\;\;\;\;\;\;\;\;\;\;\;==\sum\limits_{i=1}^3x_ia_{ii}x_i+2\mathop{\sum\limits^3\sum\limits^3}\limits_{i<j}x_ia_{ij}x_j\;對三個變量求偏微分:\) \[\frac{\partial F}{\partial x_1}=2(a_{11}x_1+a_{12}x_2+a_{13}x_3)\\ \frac{\partial F}{\partial x_2}=2(a_{12}x_1+a_{22}x_2+a_{23}x_3)\\ \frac{\partial F}{\partial x_3}=2(a_{13}x_1+a_{23}x_2+a_{33}x_3)\]

  8. 函數 \(F(x_1,x_2,x_3)=a_{11}x_1x_1+a_{12}x_1x_2+a_{13}x_1x_3 \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+a_{12}x_2x_1+a_{12}x_2x_2+a_{23}x_2x_3\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+a_{13}x_3x_1+a_{23}x_3x_2+a_{33}x_3x_3\\ G(x_1,x_2,x_3)=b_{11}x_1x_1+b_{12}x_1x_2+b_{13}x_1x_3 \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+b_{12}x_2x_1+b_{12}x_2x_2+b_{23}x_2x_3\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+b_{13}x_3x_1+b_{23}x_3x_2+b_{33}x_3x_3\\\;對三個變量求偏微分:\)

\[ \frac{\partial}{\partial x_1}(\frac{F}{G})=\frac{\frac{\partial F}{\partial x_1}G(x_1,x_2,x_3)-F(x_1,x_2,x_3)\frac{\partial G}{\partial x_1}}{\left\{G(x_1,x_2,x_3)\right\}^2}\\ =2\cdot \frac{(a_{11}x_1+a_{12}x_2+a_{13}x_3)\cdot G(x_1,x_2,x_3)-F(x_1,x_2,x_3)\cdot (b_{11}x_1+b_{12}x_2+b_{13}x_3)}{\left\{G(x_1,x_2,x_3)\right\}^2}\\ \\ \frac{\partial}{\partial x_2}(\frac{F}{G})=\frac{\frac{\partial F}{\partial x_2}G(x_1,x_2,x_3)-F(x_1,x_2,x_3)\frac{\partial G}{\partial x_2}}{\left\{G(x_1,x_2,x_3)\right\}^2}\\ =2\cdot \frac{(a_{12}x_1+a_{22}x_2+a_{23}x_3)\cdot G(x_1,x_2,x_3)-F(x_1,x_2,x_3)\cdot (b_{12}x_1+b_{22}x_2+b_{23}x_3)}{\left\{G(x_1,x_2,x_3)\right\}^2}\\ \frac{\partial}{\partial x_3}(\frac{F}{G})=\frac{\frac{\partial F}{\partial x_3}G(x_1,x_2,x_3)-F(x_1,x_2,x_3)\frac{\partial G}{\partial x_3}}{\left\{G(x_1,x_2,x_3)\right\}^2}\\ =2\cdot \frac{(a_{13}x_1+a_{23}x_2+a_{33}x_3)\cdot G(x_1,x_2,x_3)-F(x_1,x_2,x_3)\cdot (b_{13}x_1+b_{23}x_2+b_{33}x_3)}{\left\{G(x_1,x_2,x_3)\right\}^2}\\ \]

2次(2階)偏微分 【二階導數】:

函數 \(F(a_1,a_2,\cdots,a_n)\)\(a_i\) 取偏微分 \(\frac{\partial F}{\partial a_i}\) 時,記作 \(\frac{\partial^2 F}{\partial a_i^2}\) ; 取變量 \(a_j\) 的偏微分時記作 \(\frac{\partial^2 F}{\partial a_i\partial a_j}\) 或者 \(\frac{\partial^2 F}{\partial a_j\partial a_i}\)。 這些都被稱爲是函數 \(F\) 的2次(2階)偏微分。

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