导数

最新推荐文章于 2025-04-20 15:57:52 发布

原创最新推荐文章于 2025-04-20 15:57:52 发布 · 438 阅读

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本文深入探讨了向量的导数概念，包括矩阵与向量的乘法，以及导数的计算。此外，还介绍了最小二乘法在解决线性方程组问题中的应用。

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向量的导数

$A$ 为 $\times n$ 的矩阵， $x$ 为 $\times 1$ 的列向量，则 $A x$ 为 $\times 1$ 的列向量，记作 $y⃗=A⋅x⃗\vec y = A \cdot \vec x$

$\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \\ ~ \\ \vec x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \\ ~ \\ \vec y = A \cdot \vec x = \begin{bmatrix} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n \end{bmatrix}$

$\frac{\partial \vec y}{\partial \vec x} = \frac{\partial A \cdot \vec x}{\partial \vec x} = \begin{bmatrix} \frac{a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n}{x_1} & \frac{a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n}{x_1} & \cdots & \frac{a_{m1}x_1 + a_{1m2}x_2 + \cdots + a_{mn}x_1}{x_1}\\ \frac{a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n}{x_2} & \frac{a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n}{x_2} & \cdots & \frac{a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n}{x_2}\\ \vdots & \vdots & \vdots& \vdots \\ \frac{a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n}{x_n} & \frac{a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n}{x_n} & \cdots & \frac{a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n}{x_n}\\ \end{bmatrix} \\ ~ \\ = \begin{bmatrix} a_{11} & a_{21} & \cdots & a_{m1} \\ a_{12} & a_{22} & \cdots & a_{m2} \\ \vdots & \vdots & \vdots & \vdots \\ a_{1n} & a_{2n} & \cdots & a_{mn} \end{bmatrix} = A^T$

$\frac{\partial \vec y}{\partial (\vec x)^T} = \frac{\partial A \cdot \vec x}{\partial (\vec x)^T} = \begin{bmatrix} \frac{a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n}{x_1} & \frac{a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n}{x_2} & \cdots & \frac{a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_1}{x_n}\\ \frac{a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n}{x_1} & \frac{a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n}{x_2} & \cdots & \frac{a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n}{x_n}\\ \vdots & \vdots & \vdots& \vdots \\ \frac{a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n}{x_1} & \frac{a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n}{x_2} & \cdots & \frac{a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n}{x_n}\\ \end{bmatrix} \\ ~ \\ = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} = A$

${\vec x}^T \cdot A = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix} \cdot \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \\ = \begin{bmatrix} a_{11}x_1+ a_{21}x_2 + \cdots + a_{m1}x_n & a_{12}x_1+ a_{22}x_2 + \cdots + a_{m2}x_n & \cdots & a_{1n}x_1+ a_{2n}x_2 + \cdots + a_{mn}x_n \end{bmatrix} \\ ~ \\ \frac{\partial ({\vec x}^T \cdot A)}{\partial \vec x} = \begin{bmatrix} \frac{a_{11}x_1+ a_{21}x_2 + \cdots + a_{m1}x_n }{x_1} & \frac{a_{12}x_1+ a_{22}x_2 + \cdots + a_{m2}x_n}{x_1} & \cdots & \frac{a_{1n}x_1+ a_{2n}x_1 + \cdots + a_{mn}x_n}{x_1} \\ \\ \frac{a_{11}x_1+ a_{21}x_2 + \cdots + a_{m1}x_n }{x_2} & \frac{a_{12}x_1+ a_{22}x_2 + \cdots + a_{m2}x_n}{x_2} & \cdots & \frac{a_{1n}x_1+ a_{2n}x_2 + \cdots + a_{mn}x_n}{x_2} \\ \vdots & \vdots & \vdots & \vdots \\ \\ \frac{a_{11}x_1+ a_{21}x_2 + \cdots + a_{m1}x_n }{x_n} & \frac{a_{12}x_1+ a_{22}x_1 + \cdots + a_{m2}x_n}{x_n} & \cdots & \frac{a_{1n}x_1+ a_{2n}x_1 + \cdots + a_{mn}x_n}{x_n} \end{bmatrix} \\ ~ \\ =\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & \vdots & a_{mn} \end{bmatrix} = A$

$A$ 为 $\times n$ 的矩阵， $x⃗\vec x$ 为 $\times 1$ 的列向量，记 $(\vec x)^T \cdot A \cdot \vec x$
$\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \\ ~ \\ \vec x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \\ ~ \\ (\vec x)^T = \begin{bmatrix} x_1 & x_2 & \cdots & x_n\end{bmatrix} \\ ~ \\ {\vec x}^T \cdot A = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix} \cdot \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \\ = \begin{bmatrix} a_{11}x_1+ a_{21}x_2 + \cdots + a_{n1}x_n & a_{12}x_1+ a_{22}x_2 + \cdots + a_{n2}x_n & \cdots & a_{1n}x_1+ a_{2n}x_2 + \cdots + a_{nn}x_n \end{bmatrix} \\ ~ \\ (\vec x)^T \cdot A \cdot \vec x = \\ ~ \\ (a_{11}x_1+ a_{21}x_2 + \cdots + a_{n1}x_n)x_1 + (a_{12}x_1+ a_{22}x_2 + \cdots + a_{n2}x_n)x_2 + \cdots + (a_{1n}x_1+ a_{2n}x_2 + \cdots + a_{nn}x_n)x_n = y$

$\frac{\partial y}{\partial \vec x} = \begin{bmatrix} \frac{(a_{11}x_1+ a_{21}x_2 + \cdots + a_{n1}x_n)x_1 + (a_{12}x_1+ a_{22}x_2 + \cdots + a_{n2}x_n)x_2 + \cdots + (a_{1n}x_1+ a_{2n}x_2 + \cdots + a_{nn}x_n)x_n }{x_1} \\ \frac{(a_{11}x_1+ a_{21}x_2 + \cdots + a_{n1}x_n)x_1 + (a_{12}x_1+ a_{22}x_2 + \cdots + a_{n2}x_n)x_2 + \cdots + (a_{1n}x_1+ a_{2n}x_2 + \cdots + a_{nn}x_n)x_n}{x_2} \\ \vdots \\ \frac{(a_{11}x_1+ a_{21}x_2 + \cdots + a_{n1}x_n)x_1 + (a_{12}x_1+ a_{22}x_2 + \cdots + a_{n2}x_n)x_2 + \cdots + (a_{1n}x_1+ a_{2n}x_2 + \cdots + a_{nn}x_n)x_n }{x_n} \end{bmatrix} \\ ~ \\ =\begin{bmatrix} (2a_{11}x_1+ a_{21}x_2 + \cdots + a_{n1}x_n) + (a_{12}x_2) + \cdots + a_{1n}x_n \\ (a_{21}x_1) + (a_{12}x_1+ 2a_{22}x_2 + \cdots + a_{n2}x_n)x_2 + \cdots + a_{2n}x_n \\ \vdots \\ (a_{n1}x_1) +a_{n2}x_2 + \cdots + (a_{1n}x_1+ a_{2n}x_2 + \cdots + 2a_{nn}x_n) \end{bmatrix} \\ ~ \\ =\begin{bmatrix} 2a_{11}x_1+ a_{21}x_2 + (a_{12}x_2) +\cdots + a_{1n}x_n + a_{n1}x_n) \\ a_{21}x_1 +a_{12}x_1 + 2a_{22}x_2 + \cdots + a_{2n}x_n + a_{n2}x_n \\ \vdots \\ (a_{n1}x_1) +a_{1n}x_1 + a_{n2}x_2 + a_{2n}x_2 +\cdots + 2a_{nn}x_n) \end{bmatrix} \\ ~ \\ ==\begin{bmatrix} 2a_{11}x_1+ (a_{21} + a_{12})x_2 +\cdots + (a_{1n} + a_{n1})x_n) \\ (a_{21} +a_{12})x_1 + 2a_{22}x_2 + \cdots + (a_{2n} + a_{n2})x_n \\ \vdots \\ (a_{n1}+a_{1n})x_1 + (a_{n2}+ a_{2n})x_2 +\cdots + 2a_{nn}x_n) \end{bmatrix} \\ ~ \\ A^T + A =\begin{bmatrix} a_{11} & a_{21} & \cdots & a_{n1} \\ a_{12} & a_{22} & \cdots & a_{n2} \\ \vdots & \vdots & \vdots & \vdots \\ a_{1n} & a_{2n} & \cdots & a_{nn} \end{bmatrix} + \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \\~ \\ = \begin{bmatrix} 2a_{11} & a_{12} + a_{21} & \cdots & a_{1n} + a_{n1}\\ a_{21} + a_{21}& 2a_{22} & \cdots & a_{2n} + a_{n2}\\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} + a_{n1}& a_{n2} + a_{2n}& \cdots & 2a_{nn} \end{bmatrix} \\ ~ \\ (A^T + A) \cdot \vec x = \begin{bmatrix} 2a_{11} x_1 + (a_{12} + a_{21})x_2 + \cdots + (a_{1n} + a_{n1})x_n\\ (a_{21} +a_{21})x_1+ 2a_{22} x_2+ \cdots +( a_{2n} + a_{n2})x_n\\ \vdots \\ (a_{n1} + a_{n1})x_1+ (a_{n2} + a_{2n})x_2 +\cdots + 2a_{nn}x_n \end{bmatrix} \\ ~ \\ \therefore \frac{\partial y}{\partial \vec x} = (A^T + A) \cdot \vec x$