目錄

• 線性代數
  • 一、基本知識
  • 二、向量操作
  • 三、矩陣運算

線性代數

一、基本知識

1. 本書中所有的向量都是列向量的形式：
   \[\mathbf{\vec x}=(x_1,x_2,\cdots,x_n)^T=\begin{bmatrix}x_1\\x_2\\ \vdots \\x_n\end{bmatrix}\] 本書中所有的矩 \(\mathbf X\in \mathbb R^{m\times n}\) 都表示為：
   \[\mathbf X = \begin{bmatrix} x_{1,1}&x_{1,2}&\cdots&x_{1,n}\\ x_{2,1}&x_{2,2}&\cdots&x_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m,1}&x_{m,2}&\cdots&x_{m,n}\\ \end{bmatrix}\]
   簡寫為 \((x_{i,j})_{m\times n}\) 或 \([x_{i,j}]_{m\times n}\) 。
2. 矩陣的F範數：設矩 \(\mathbf A=(a_{i,j})_{m\times n}\) ，則其F範數為 \(||\mathbf A||_F=\sqrt{\sum_{i,j}a_{i,j}^{2}}\) 。
   它是向量 \(L_2\) 範數的推廣。
3. 矩陣的跡：設矩 \(\mathbf A=(a_{i,j})_{m\times n}\) ， $ \mathbf A$ 的跡為 \(tr(\mathbf A)=\sum_{i}a_{i,i}\) 。
   跡的性質有：
   • \(\mathbf A\)
     的F 範數等 \(\mathbf A\mathbf A^T\) 的跡的平方根 \(||\mathbf A||_F=\sqrt{tr(\mathbf A \mathbf A^{T})}\) 。
   • \(\mathbf A\) 的跡等 \(\mathbf A^T\) 的跡 \(tr(\mathbf A)=tr(\mathbf A^{T})\) 。
   • 交換律：假設 \(\mathbf A\in \mathbb R^{m\times n},\mathbf B\in \mathbb R^{n\times m}\) ，則有 \(tr(\mathbf A\mathbf B)=tr(\mathbf B\mathbf A)\)
     。
   • 結合律 \(tr(\mathbf A\mathbf B\mathbf C)=tr(\mathbf C\mathbf A\mathbf B)=tr(\mathbf B\mathbf C\mathbf A)\) 。

二、向量操作

1. 一組向 \(\mathbf{\vec v}_1,\mathbf{\vec v}_2,\cdots,\mathbf{\vec v}_n\) 是線性相關的：指存在一組不全為零的實 \(a_1,a_2,\cdots,a_n\) ，使得 \(\sum_{i=1}^{n}a_i\mathbf{\vec v}_i=\mathbf{\vec 0}\) 。
   一組向 \(\mathbf{\vec v}_1,\mathbf{\vec v}_2,\cdots,\mathbf{\vec v}_n\) 是線性無關的，當且僅 \(a_i=0,i=1,2,\cdots,n\) 時，才有 \(\sum_{i=1}^{n}a_i\mathbf{\vec v}_i=\mathbf{\vec 0}\) 。
2. 一個向量空間所包含的最大線性無關向量的數目，稱作該向量空間的維數。

3. 三維向量的點積 \(\mathbf{\vec u}\cdot\mathbf{\vec v} =u _xv_x+u_yv_y+u_zv_z = |\mathbf{\vec u}| | \mathbf{\vec v}| \cos(\mathbf{\vec u},\mathbf{\vec v})\) 。
   

   

4. 三維向量的叉積：
   \[\mathbf{\vec w}=\mathbf{\vec u}\times \mathbf{\vec v}=\begin{bmatrix}\mathbf{\vec i}& \mathbf{\vec j}&\mathbf{\vec k}\\ u_x&u_y&u_z\\ v_x&v_y&v_z\\ \end{bmatrix}\] 其 \(\mathbf{\vec i}, \mathbf{\vec j},\mathbf{\vec k}\) 分別 \(x,y,z\) 軸的單位向量。
   \[\mathbf{\vec u}=u_x\mathbf{\vec i}+u_y\mathbf{\vec j}+u_z\mathbf{\vec k},\quad \mathbf{\vec v}=v_x\mathbf{\vec i}+v_y\mathbf{\vec j}+v_z\mathbf{\vec k}\] ​
   • $\mathbf{\vec u} $ 和 \(\mathbf{\vec v}\) 的叉積垂直於 \(\mathbf{\vec u},\mathbf{\vec v}\) 構成的平面，其方向符合右手規則。
   • 叉積的模等於 \(\mathbf{\vec u},\mathbf{\vec v}\) 構成的平行四邊形的面積
   • \(\mathbf{\vec u}\times \mathbf{\vec v}=-\mathbf{\vec v}\times \mathbf{\vec u}\)
   • $\mathbf{\vec u}\times( \mathbf{\vec v} \times \mathbf{\vec w})=(\mathbf{\vec u}\cdot \mathbf{\vec w})\mathbf{\vec v}-(\mathbf{\vec u}\cdot \mathbf{\vec v})\mathbf{\vec w} $
     
5. 三維向量的混合積：
   \[[\mathbf{\vec u} \;\mathbf{\vec v} \;\mathbf{\vec w}]=(\mathbf{\vec u}\times \mathbf{\vec v})\cdot \mathbf{\vec w}= \mathbf{\vec u}\cdot (\mathbf{\vec v} \times \mathbf{\vec w})\\ =\begin{vmatrix} u_x&u_y&u_z\\ v_x&v_y&v_z\\ w_x&w_y&w_z \end{vmatrix} =\begin{vmatrix} u_x&v_x&w_x\\ u_y&v_y&w_y\\ u_z&v_z&w_z\end{vmatrix} \] 其物理意義為： \(\mathbf{\vec u} ,\mathbf{\vec v} ,\mathbf{\vec w}\) 為三個稜邊所圍成的平行六面體的體積。 \(\mathbf{\vec u} ,\mathbf{\vec v} ,\mathbf{\vec w}\) 構成右手系時，該平行六面體的體積為正號。

6. 兩個向量的並矢：給定兩個向 \(\mathbf {\vec x}=(x_1,x_2,\cdots,x_n)^{T}, \mathbf {\vec y}= (y_1,y_2,\cdots,y_m)^{T}\) ，則向量的並矢記作：
   \[\mathbf {\vec x}\mathbf {\vec y} =\begin{bmatrix}x_1y_1&x_1y_2&\cdots&x_1y_m\\ x_2y_1&x_2y_2&\cdots&x_2y_m\\ \vdots&\vdots&\ddots&\vdots\\ x_ny_1&x_ny_2&\cdots&x_ny_m\\ \end{bmatrix}\] 也記 \(\mathbf {\vec x}\otimes\mathbf {\vec y}\) 或 \(\mathbf {\vec x} \mathbf {\vec y}^{T}\) 。

三、矩陣運算

1. 給定兩個矩 \(\mathbf A=(a_{i,j}) \in \mathbb R^{m\times n},\mathbf B=(b_{i,j}) \in \mathbb R^{m\times n}\) ，定義：
   • 阿達馬積Hadamard product（又稱作逐元素積）：
     \[\mathbf A \circ \mathbf B =\begin{bmatrix} a_{1,1}b_{1,1}&a_{1,2}b_{1,2}&\cdots&a_{1,n}b_{1,n}\\ a_{2,1}b_{2,1}&a_{2,2}b_{2,2}&\cdots&a_{2,n}b_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m,1}b_{m,1}&a_{m,2}b_{m,2}&\cdots&a_{m,n}b_{m,n}\end{bmatrix}\]
   • 克羅內積Kronnecker product：
     \[\mathbf A \otimes \mathbf B =\begin{bmatrix}a_{1,1}\mathbf B&a_{1,2}\mathbf B&\cdots&a_{1,n}\mathbf B\\ a_{2,1}\mathbf B&a_{2,2}\mathbf B&\cdots&a_{2,n}\mathbf B\\ \vdots&\vdots&\ddots&\vdots\\ a_{m,1}\mathbf B&a_{m,2}\mathbf B&\cdots&a_{m,n}\mathbf B \end{bmatrix}\]

2. \(\mathbf {\vec x},\mathbf {\vec a},\mathbf {\vec b},\mathbf {\vec c}\) \(n\) 階向量 \(\mathbf A,\mathbf B,\mathbf C,\mathbf X\) \(n\) 階方陣，則有：
   \[\frac{\partial(\mathbf {\vec a}^{T}\mathbf {\vec x}) }{\partial \mathbf {\vec x} }=\frac{\partial(\mathbf {\vec x}^{T}\mathbf {\vec a}) }{\partial \mathbf {\vec x} } =\mathbf {\vec a}\] \[\frac{\partial(\mathbf {\vec a}^{T}\mathbf X\mathbf {\vec b}) }{\partial \mathbf X }=\mathbf {\vec a}\mathbf {\vec b}^{T}=\mathbf {\vec a}\otimes\mathbf {\vec b}\in \mathbb R^{n\times n}\] \[\frac{\partial(\mathbf {\vec a}^{T}\mathbf X^{T}\mathbf {\vec b}) }{\partial \mathbf X }=\mathbf {\vec b}\mathbf {\vec a}^{T}=\mathbf {\vec b}\otimes\mathbf {\vec a}\in \mathbb R^{n\times n}\] \[\frac{\partial(\mathbf {\vec a}^{T}\mathbf X\mathbf {\vec a}) }{\partial \mathbf X }=\frac{\partial(\mathbf {\vec a}^{T}\mathbf X^{T}\mathbf {\vec a}) }{\partial \mathbf X }=\mathbf {\vec a}\otimes\mathbf {\vec a}\] \[\frac{\partial(\mathbf {\vec a}^{T}\mathbf X^{T}\mathbf X\mathbf {\vec b}) }{\partial \mathbf X }=\mathbf X(\mathbf {\vec a}\otimes\mathbf {\vec b}+\mathbf {\vec b}\otimes\mathbf {\vec a})\] \[\frac{\partial[(\mathbf A\mathbf {\vec x}+\mathbf {\vec a})^{T}\mathbf C(\mathbf B\mathbf {\vec x}+\mathbf {\vec b})]}{\partial \mathbf {\vec x}}=\mathbf A^{T}\mathbf C(\mathbf B\mathbf {\vec x}+\mathbf {\vec b})+\mathbf B^{T}\mathbf C(\mathbf A\mathbf {\vec x}+\mathbf {\vec a})\] \[\frac{\partial (\mathbf {\vec x}^{T}\mathbf A \mathbf {\vec x})}{\partial \mathbf {\vec x}}=(\mathbf A+\mathbf A^{T})\mathbf {\vec x}\] \[\frac{\partial[(\mathbf X\mathbf {\vec b}+\mathbf {\vec c})^{T}\mathbf A(\mathbf X\mathbf {\vec b}+\mathbf {\vec c})]}{\partial \mathbf X}=(\mathbf A+\mathbf A^{T})(\mathbf X\mathbf {\vec b}+\mathbf {\vec c})\mathbf {\vec b}^{T} \] \[\frac{\partial (\mathbf {\vec b}^{T}\mathbf X^{T}\mathbf A \mathbf X\mathbf {\vec c})}{\partial \mathbf X}=\mathbf A^{T}\mathbf X\mathbf {\vec b}\mathbf {\vec c}^{T}+\mathbf A\mathbf X\mathbf {\vec c}\mathbf {\vec b}^{T}\]

3. 如 \(f\) 是一元函式，則：
   • 其逐元向量函式為 \(f(\mathbf{\vec x}) =(f(x_1),f(x_2),\cdots,f(x_n))^{T}\) 。
   • 其逐矩陣函式為：
     \[f(\mathbf X)=\begin{bmatrix} f(x_{1,1})&f(x_{1,2})&\cdots&f(x_{1,n})\\ f(x_{2,1})&f(x_{2,2})&\cdots&f(x_{2,n})\\ \vdots&\vdots&\ddots&\vdots\\ f(x_{m,1})&f(x_{m,2})&\cdots&f(x_{m,n})\\ \end{bmatrix}\]
   • 其逐元導數分別為：
     \[f^{\prime}(\mathbf{\vec x}) =(f^{\prime}(x1),f^{\prime}(x2),\cdots,f^{\prime}(x_n))^{T}\\ f^{\prime}(\mathbf X)=\begin{bmatrix} f^{\prime}(x_{1,1})&f^{\prime}(x_{1,2})&\cdots&f^{\prime}(x_{1,n})\\ f^{\prime}(x_{2,1})&f^{\prime}(x_{2,2})&\cdots&f^{\prime}(x_{2,n})\\ \vdots&\vdots&\ddots&\vdots\\ f^{\prime}(x_{m,1})&f^{\prime}(x_{m,2})&\cdots&f^{\prime}(x_{m,n})\\ \end{bmatrix}\]
4. 各種型別的偏導數：
   • 標量對標量的偏導數 \(\frac{\partial u}{\partial v}\) 。
   • 標量對向量 \(n\) 維向量）的偏導數 \(\frac{\partial u}{\partial \mathbf {\vec v}}=(\frac{\partial u}{\partial v_1},\frac{\partial u}{\partial v_2},\cdots,\frac{\partial u}{\partial v_n})^{T}\) 。
   • 標量對矩陣 \(m\times n\) 階矩陣)的偏導數：
     \[\frac{\partial u}{\partial \mathbf V}=\begin{bmatrix} \frac{\partial u}{\partial V_{1,1}}&\frac{\partial u}{\partial V_{1,2}}&\cdots&\frac{\partial u}{\partial V_{1,n}}\\ \frac{\partial u}{\partial V_{2,1}}&\frac{\partial u}{\partial V_{2,2}}&\cdots&\frac{\partial u}{\partial V_{2,n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial u}{\partial V_{m,1}}&\frac{\partial u}{\partial V_{m,2}}&\cdots&\frac{\partial u}{\partial V_{m,n}} \end{bmatrix}\]
   • 向量 \(m\) 維向量）對標量的偏導數 \(\frac{\partial \mathbf {\vec u}}{\partial v}=(\frac{\partial u_1}{\partial v},\frac{\partial u_2}{\partial v},\cdots,\frac{\partial u_m}{\partial v})^{T}\) 。
   • 向量 \(m\) 維向量）對向量 \(n\) 維向量)的偏導數（雅可比矩陣，行優先）
     \[\frac{\partial \mathbf {\vec u}}{\partial \mathbf {\vec v}}=\begin{bmatrix} \frac{\partial u_1}{\partial v_1}&\frac{\partial u_1}{\partial v_2}&\cdots&\frac{\partial u_1}{\partial v_n}\\ \frac{\partial u_2}{\partial v_1}&\frac{\partial u_2}{\partial v_2}&\cdots&\frac{\partial u_2}{\partial v_n}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial u_m}{\partial v_1}&\frac{\partial u_m}{\partial v_2}&\cdots&\frac{\partial u_m}{\partial v_n} \end{bmatrix}\] 如果為列優先，則為上面矩陣的轉置。
   • 矩陣 \(m\times n\) 階矩陣)對標量的偏導數
     \[\frac{\partial \mathbf U}{\partial v}=\begin{bmatrix} \frac{\partial U_{1,1}}{\partial v}&\frac{\partial U_{1,2}}{\partial v}&\cdots&\frac{\partial U_{1,n}}{\partial v}\\ \frac{\partial U_{2,1}}{\partial v}&\frac{\partial U_{2,2}}{\partial v}&\cdots&\frac{\partial U_{2,n}}{\partial v}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial U_{m,1}}{\partial v}&\frac{\partial U_{m,2}}{\partial v}&\cdots&\frac{\partial U_{m,n}}{\partial v} \end{bmatrix}\]
5. 對於矩陣的跡，有下列偏導數成立：
   \[\frac{\partial [tr(f(\mathbf X))]}{\partial \mathbf X }=(f^{\prime}(\mathbf X))^{T}\] \[\frac{\partial [tr(\mathbf A\mathbf X\mathbf B)]}{\partial \mathbf X }=\mathbf A^{T}\mathbf B^{T} \] \[\frac{\partial [tr(\mathbf A\mathbf X^{T}\mathbf B)]}{\partial \mathbf X }=\mathbf B\mathbf A \] \[\frac{\partial [tr(\mathbf A\otimes\mathbf X )]}{\partial \mathbf X }=tr(\mathbf A)\mathbf I\] \[\frac{\partial [tr(\mathbf A\mathbf X \mathbf B\mathbf X)]}{\partial \mathbf X }=\mathbf A^{T}\mathbf X^{T}\mathbf B^{T}+\mathbf B^{T}\mathbf X \mathbf A^{T} \] \[\frac{\partial [tr(\mathbf X^{T} \mathbf B\mathbf X \mathbf C)]}{\partial \mathbf X }=(\mathbf B^{T}+\mathbf B)\mathbf X \mathbf C \mathbf C^{T} \] \[\frac{\partial [tr(\mathbf C^{T}\mathbf X^{T} \mathbf B\mathbf X \mathbf C)]}{\partial \mathbf X }=\mathbf B\mathbf X \mathbf C +\mathbf B^{T}\mathbf X \mathbf C^{T} \] \[\frac{\partial [tr(\mathbf A\mathbf X \mathbf B\mathbf X^{T} \mathbf C)]}{\partial \mathbf X }= \mathbf A^{T}\mathbf C^{T}\mathbf X\mathbf B^{T}+\mathbf C \mathbf A \mathbf X \mathbf B\] \[\frac{\partial [tr((\mathbf A\mathbf X\mathbf B+\mathbf C)(\mathbf A\mathbf X\mathbf B+\mathbf C))]}{\partial \mathbf X }= 2\mathbf A ^{T}(\mathbf A\mathbf X\mathbf B+\mathbf C)\mathbf B^{T}\]

6. 假 \(\mathbf U= f(\mathbf X)\) 是關 \(\mathbf X\) 的矩陣值函式 \(f:\mathbb R^{m\times n}\rightarrow \mathbb R^{m\times n}\) ）， \(g(\mathbf U)\) 是關 \(\mathbf U\) 的實值函式 $g:\mathbb R^{m\times n}\rightarrow \mathbb R $ ），則下面鏈式法則成立：
   \[\frac{\partial g(\mathbf U)}{\partial \mathbf X}= \left(\frac{\partial g(\mathbf U)}{\partial x_{i,j}}\right)_{m\times n}=\begin{bmatrix} \frac{\partial g(\mathbf U)}{\partial x_{1,1}}&\frac{\partial g(\mathbf U)}{\partial x_{1,2}}&\cdots&\frac{\partial g(\mathbf U)}{\partial x_{1,n}}\\ \frac{\partial g(\mathbf U)}{\partial x_{2,1}}&\frac{\partial g(\mathbf U)}{\partial x_{2,2}}&\cdots&\frac{\partial g(\mathbf U)}{\partial x_{2,n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial g(\mathbf U)}{\partial x_{m,1}}&\frac{\partial g(\mathbf U)}{\partial x_{m,2}}&\cdots&\frac{\partial g(\mathbf U)}{\partial x_{m,n}}\\ \end{bmatrix}\\ =\left(\sum_{k}\sum_{l}\frac{\partial g(\mathbf U)}{\partial u_{k,l}}\frac{\partial u_{k,l}}{\partial x_{i,j}}\right)_{m\times n}=\left(tr\left[\left(\frac{\partial g(\mathbf U)}{\partial \mathbf U}\right)^{T}\frac{\partial \mathbf U}{\partial x_{i,j}}\right]\right)_{m\times n}\]

本文轉載自華校專老師部落格，部落格地址：http://www.huaxiaozhuan.com/