Jacobian

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最新推荐文章于 2024-03-05 10:12:17 发布

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文章标签： foundation

本文详细介绍了雅可比矩阵的概念及其在多变量函数中的应用。包括雅可比矩阵的定义、如何通过偏导数构建矩阵，以及其行列式的含义。

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Given a set $\mathbf{y=f(x)}$ of $n$ variables $x_1,...,x_n$ , written explicitly as

y = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ f 1 (x) f 2 (x) ⋮ f n (x) ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥

$\mathbf y = \left[ \begin{array}{c} f_1(\mathbf x)\\ f_2(\mathbf x)\\ \vdots\\ f_n(\mathbf x) \end{array} \right]$
or more explicitly as

⎧ ⎩ ⎨ ⎪ ⎪ y 1 = f 1 (x 1, . . ., x n) ⋮ y n = f n (x 1, . . ., x n)

$\begin{cases} y_1=f_1(x_1,...,x_n)\\ \vdots\\ y_n=f_n(x_1,...,x_n)\\ \end{cases}$
the Jacobian matrix, sometimes simply called “the Jacobian”(Simon and Blume 1994) is defined by

J (x 1, . . ., x n) = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ \partial y 1 \partial x 1 ⋮ \partial y n \partial x 1 \dots ⋱ \dots \partial y 1 \partial x n ⋮ \partial y n \partial x n ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

$J(x_1,...,x_n)= \begin{bmatrix} \frac{\partial y_1}{\partial x_1}&\cdots&\frac{\partial y_1}{\partial x_n}\\ \vdots&\ddots&\vdots\\ \frac{\partial y_n}{\partial x_1}&\cdots&\frac{\partial y_n}{\partial x_n} \end{bmatrix}$
The determinant of J is the Jacobian determinant (confusingly, often called “the Jacobian” as well) and is denoted

J = ∣ ∣ ∣ \partial ( y 1 , . . . , y n ) \partial ( x 1 , . . . , x n ) ∣ ∣ ∣

$J=\begin{vmatrix} \frac{\partial (y_1,...,y_n)}{\partial (x_1,...,x_n)} \end{vmatrix}$

Practical 4

The Jacobian is a $m×n$ matrix of derivatives for a multivariate function $f:R^n \to R^m$ :

d f d x = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ \partial f 1 \partial x 1 ⋮ \partial f m \partial x 1 \dots ⋱ \dots \partial f 1 \partial x n ⋮ \partial f m \partial x n ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

$\frac{d\mathbf f}{d\mathbf x}= \begin{bmatrix} \frac{\partial f_1}{\partial x_1}&\cdots&\frac{\partial f_1}{\partial x_n}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_m}{\partial x_1}&\cdots&\frac{\partial f_m}{\partial x_n} \end{bmatrix}$
each

i $i$ th row being a gradient of an element,

fi $f_i$ , of the output vector,

f $\mathbf f$ .