矩阵求导公式的数学推导

2022-01-19

字数统计: 194字 | 阅读时长≈ 1分钟

矩阵求导公式推导

公式一

$\frac{\partial( \pmb{x}^T \pmb{a})}{\partial{\pmb{x}}} = \frac{\partial( \pmb{a}^T\pmb{x})}{\partial{\pmb{x}}} = \pmb{a} \\\\$

其中，α为常数列向量。

证明：

$\frac{\partial( \pmb{x}^T \pmb{a})}{\partial{\pmb{x}}} = \frac{\partial( \pmb{a}^T\pmb{x})}{\partial{\pmb{x}}} \\\\ = \frac{\partial( a_1x_1+a_2x_2+\cdots+a_nx_n)}{\partial{\pmb{x}}} \\\\ = \begin{bmatrix} \frac{\partial( a_1x_1+a_2x_2+\cdots+a_nx_n)}{\partial{x_1}} \\ \frac{\partial( a_1x_1+a_2x_2+\cdots+a_nx_n)}{\partial{x_2}} \\ \vdots \\ \frac{\partial( a_1x_1+a_2x_2+\cdots+a_nx_n)}{\partial{x_n}} \end{bmatrix} \\\\ = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix} \\\\ = \pmb{a}$

公式二

$\frac{\partial( \pmb{x}^T \pmb{A}\pmb{x})}{\partial{\pmb{x}}} = (\pmb{A}+\pmb{A}^T) \pmb{x}$

其中，A是常数矩阵，

$\pmb{A}_{n \times n}=(a_{ij})_{i=1,j=1}^{n,n}$

证明：

$\begin{align} \frac{\partial( \pmb{x}^T \pmb{A}\pmb{x})}{\partial{\pmb{x}}} &= \frac{\partial(a_{11}x_1x_1+a_{12}x_1x_2+\cdots+a_{1n}x_1x_n \\ +a_{21}x_2x_1+a_{22}x_2x_2+\cdots+a_{2n}x_2x_n \\ + \cdots \\ +a_{n1}x_nx_1+a_{n2}x_nx_2+\cdots+a_{nn}x_nx_n)}{\partial{\pmb{x}}} \\\\ &= \begin{bmatrix} \frac{\partial(a_{11}x_1x_1+a_{12}x_1x_2+\cdots+a_{1n}x_1x_n \\ +a_{21}x_2x_1+a_{22}x_2x_2+\cdots+a_{2n}x_2x_n \\ + \cdots \\ +a_{n1}x_nx_1+a_{n2}x_nx_2+\cdots+a_{nn}x_nx_n)}{\partial{x_1}} \\ \frac{\partial(a_{11}x_1x_1+a_{12}x_1x_2+\cdots+a_{1n}x_1x_n \\ +a_{21}x_2x_1+a_{22}x_2x_2+\cdots+a_{2n}x_2x_n \\ + \cdots \\ +a_{n1}x_nx_1+a_{n2}x_nx_2+\cdots+a_{nn}x_nx_n)}{\partial{x_2}} \\ \vdots \\ \frac{\partial(a_{11}x_1x_1+a_{12}x_1x_2+\cdots+a_{1n}x_1x_n \\ +a_{21}x_2x_1+a_{22}x_2x_2+\cdots+a_{2n}x_2x_n \\ + \cdots \\ +a_{n1}x_nx_1+a_{n2}x_nx_2+\cdots+a_{nn}x_nx_n)}{\partial{x_n}} \end{bmatrix} \\\\ &= \begin{bmatrix} (a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n)+(a_{11}x_1+a_{21}x_2+\cdots+a_{n1}x_n) \\ (a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n)+(a_{12}x_1+a_{22}x_2+\cdots+a_{n2}x_n) \\ \vdots \\ (a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n)+(a_{1n}x_1+a_{2n}x_2+\cdots+a_{nn}x_n) \end{bmatrix} \\\\ &= \begin{bmatrix} a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n \\ a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n \\ \vdots \\ a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n \end{bmatrix} +\begin{bmatrix} a_{11}x_1+a_{21}x_2+\cdots+a_{n1}x_n \\ a_{12}x_1+a_{22}x_2+\cdots+a_{n2}x_n \\ \vdots \\ a_{1n}x_1+a_{2n}x_2+\cdots+a_{nn}x_n \end{bmatrix} \\\\ &= \begin{bmatrix} a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn} \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix} +\begin{bmatrix} a_{11}&a_{21}&\cdots&a_{n1}\\ a_{12}&a_{22}&\cdots&a_{n2}\\ \vdots&\vdots&\ddots&\vdots\\ a_{1n}&a_{2n}&\cdots&a_{nn} \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix} \\\\ &= \pmb{A}\pmb{x}+\pmb{A}^T \pmb{x} \end{align} \\\\ \tag{14}$