# 认识Jacobian Matrix

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In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature.

Jacobian Matrix是一个包含偏导数的矩阵。当Jacobian Matrix是一个方阵的时候，Jacobian这个词单独还可以表示此方阵的determinant。（神经网络的输出层函多个输出神经元的时候，整个网络就可以看做是一个vector function）

The matrix will contain all partial derivatives of a vector function. The main use of Jacobian is found in the transformation of coordinates.

The term Jacobian often represents both the jacobian matrix and determinants, which is defined for the finite number of function with the same number of variables. Here, each row consists of the first partial derivative of the same function, with respect to all the variables. The jacobian matrix can be of any form. It may be a square matrix or the rectangular matrix.

举个栗子：

Suppose $$f$$ : $$R^n\rightarrow R^m$$ is a function such that each of its first-order partial derivatives exist on $$R^n$$

设方程$$f$$，n个变量，m个输出，其Jacobian Matrix为：

$$J = \begin{bmatrix}\cfrac{\partial{f}}{\partial x_1}&\cfrac{\partial{f}}{\partial x_2}&\cdots&\cfrac{\partial{f}}{\partial x_n}\end{bmatrix}=\begin{bmatrix}\nabla^T f_1\\\nabla^T f_2\\\vdots\\\nabla^T f_m\end{bmatrix}=\begin{bmatrix}\cfrac{\partial{f_1}}{\partial x_1}&\cfrac{\partial{f_1}}{\partial x_2}&\cdots&\cfrac{\partial{f_1}}{\partial x_n}\\ \cfrac{\partial{f_2}}{\partial x_2}&\cfrac{\partial{f_2}}{\partial x_2}&\cdots&\cfrac{\partial{f_2}}{\partial x_n}\\ \vdots&\vdots&\ddots&\vdots\\ \cfrac{\partial{f_m}}{\partial x_2}&\cfrac{\partial{f_m}}{\partial x_2}&\cdots&\cfrac{\partial{f_m}}{\partial x_n}\end{bmatrix}_{m\times n}$$

$$m\times n$$的Jacobian Matrix，m个function，共用n个variables。

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