AI study notes[4]

theorem of implicit function

1. there are a system of m equations in n variables:

$f_i(x)=0,i=0,1,...,m$
Can the m variables,such as $x_1,x_2,...,x_m$, be defined as implicit functions of the other n-m variables such as $x_{m+1},...,x_n$?
can the system determines m variables ?

2. let $x_0\in R^n$ and satisfy the following conditions:

• $f_i(x_0)=0$
• the function $f_i\in C^1(i=1,...,m)$ in a certain neighborhood of $x_0$
• on condition that the Jacobian matrix of the system about variables such as $x_1,x_2,...,x_m$
  J f = ∂ f ∂ x = ( ∂ f 1 ( x 0 ) ∂ x 1 ⋯ ∂ f 1 ( x 0 ) ∂ x n ⋮ ⋱ ⋮ ∂ f m ( x 0 ) ∂ x 1 ⋯ ∂ f m ( x 0 ) ∂ x n ) ≠ 0 J_{\mathbf{f}} = \frac{\partial \mathbf{f}}{\partial \mathbf{x}} = \begin{pmatrix} \frac{\partial f_1(x_0)}{\partial x_1} & \cdots & \frac{\partial f_1(x_0)}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m(x_0)}{\partial x_1} & \cdots & \frac{\partial f_m(x_0)}{\partial x_n} \end{pmatrix} \neq 0 Jf​=∂x∂f​= ​∂x1​∂f1​(x0​)​⋮∂x1​∂fm​(x0​)​​⋯⋱⋯​∂xn​∂f1​(x0​)​⋮∂xn​∂fm​(x0​)​​ ​=0
  ， a neighborhood of ${\bar{x}}_0=({x_{m+1}}_0,..,{x_n}_0)\in R^{n-m}$ exists leaded to having function $g_i(\bar{x})(i=1,2,....m)$ for $\bar{x}=x_{m+1}...,x_n$in the neighborhood of ${\bar{x}}_0$ . the function of $g_i(\bar{x})(i=1,2,....m)$ satisfy The following points:
  • $g_i(\bar{x})(i=1,2,....m)\in C^1$，one continuous function.
  • ${x_i}_0=g_i({\bar{x}}_0)(i=1,2,....m)$
  • $f_i(g_1(\bar{x}),...,g_m(\bar{x}),\bar{x})=0(i=1,2,....m)$

Given a function $\mathbf{f}:{\mathbb{R}}^n\to {\mathbb{R}}^m$, where:
f ( x ) = ( f 1 ( x 1 , … , x n ) ⋮ f m ( x 1 , … , x n ) ) , \mathbf{f}(\mathbf{x}) = \begin{pmatrix} f_1(x_1, \dots, x_n) \\ \vdots \\ f_m(x_1, \dots, x_n) \end{pmatrix}, f(x)= ​f1​(x1​,…,xn​)⋮fm​(x1​,…,xn​)​ ​, the Jacobian
matrix $J_{\mathbf{F}}$ is an $m\times n$ matrix defined as: J F = ∂ F ∂ x = ( ∂ F 1 ∂ x 1 ⋯ ∂ F 1 ∂ x n ⋮ ⋱ ⋮ ∂ F m ∂ x 1 ⋯ ∂ F m ∂ x n ) . J_{\mathbf{F}} = \frac{\partial \mathbf{F}}{\partial \mathbf{x}} = \begin{pmatrix} \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{pmatrix}. JF​=∂x∂F​= ​∂x1​∂F1​​⋮∂x1​∂Fm​​​⋯⋱⋯​∂xn​∂F1​​⋮∂xn​∂Fm​​​ ​.

references

《最优化理论与算法》