Opencv 数学基础--范数(17)

╰+攻爆jí腚メ 2022-12-14 12:23 111阅读 0赞

原文地址:[https://blog.csdn.net/zaishuiyifangxym/article/details/81673491][https_blog.csdn.net_zaishuiyifangxym_article_details_81673491]

## **1、向量的范数** ##

**  向量的1-范数： ![\{\\left\\| X \\right\\|\_1\} = \\sum\\limits\_\{i = 1\}^n \{\\left| \{\{x\_i\}\} \\right|\}][left_ X _right_1_ _ _sum_limits_i _ 1_n _left_ _x_i_ _right] ; **各个元素的绝对值之和；

**  向量的2-范数：![\{\\left\\| X \\right\\|\_2\} = \{\\left( \{\\sum\\limits\_\{i = 1\}^n \{\{x\_i\}^2\} \} \\right)^\{\\frac\{1\}\{2\}\}\} = \\sqrt \{\\sum\\limits\_\{i = 1\}^n \{\{x\_i\}^2\} \}][left_ X _right_2_ _ _left_ _sum_limits_i _ 1_n _x_i_2_ _ _right_frac_1_2_ _ _sqrt _sum_limits_i _ 1_n _x_i_2_]**；每个元素的平方和再开平方根；

**  向量的无穷范数：**![\{\\left\\| X \\right\\|\_\\infty \} = \\mathop \{\\max \}\\limits\_\{1 \\le i \\le n\} \\left| \{\{x\_i\}\} \\right|][left_ X _right_infty _ _ _mathop _max _limits_1 _le i _le n_ _left_ _x_i_ _right]

***  p*\-范数：**![\{\\left\\| X \\right\\|\_p\} = \{\\left( \{\\sum\\limits\_\{i = 1\}^n \{\{\{\\left| \{\{x\_i\}\} \\right|\}^p\}\} \} \\right)^\{\\frac\{1\}\{p\}\}\}][left_ X _right_p_ _ _left_ _sum_limits_i _ 1_n _left_ _x_i_ _right_p_ _ _right_frac_1_p]，其中正整数*p*≥1，并且有![\\mathop \{\\lim \}\\limits\_\{p \\to \\infty \} \{\\left\\| X \\right\\|\_p\} = \\mathop \{\\max \}\\limits\_\{1 \\le i \\le n\} \\left| \{\{x\_i\}\} \\right|][mathop _lim _limits_p _to _infty _ _left_ X _right_p_ _ _mathop _max _limits_1 _le i _le n_ _left_ _x_i_ _right]

例：向量*X*=\[2, 3, -5, -7\] ，求向量的1-范数，2-范数和无穷范数。

向量的1-范数：各个元素的绝对值之和；![\{\\left\\| X \\right\\|\_1\}][left_ X _right_1]=2+3+5+7=17；

Matlab代码：*X*=\[2, 3, -5, -7\]; XLfs1=norm(*X*,1);

向量的2-范数：每个元素的平方和再开平方根；![\{\\left\\| X \\right\\|\_2\} = \{\\left( \{\{\\rm\{2\}\} \\times \{\\rm\{2\}\} + \{\\rm\{3\}\} \\times \{\\rm\{3\}\} + \{\\rm\{5\}\} \\times \{\\rm\{5\}\} + \{\\rm\{7\}\} \\times \{\\rm\{7\}\}\} \\right)^\{\\frac\{1\}\{2\}\}\} = 9.3274][left_ X _right_2_ _ _left_ _rm_2_ _times _rm_2_ _ _rm_3_ _times _rm_3_ _ _rm_5_ _times _rm_5_ _ _rm_7_ _times _rm_7_ _right_frac_1_2_ _ 9.3274]；

Matlab代码：*X*=\[2, 3, -5, -7\]; XLfs2=norm(X,2);

向量的无穷范数：

（1）正无穷范数：向量的所有元素的绝对值中**最大**的；即*X*的正无穷范数为：7；

Matlab代码：*X*=\[2, 3, -5, -7\]; XLfsz=norm(*X*,inf);

（2）负无穷范数：向量的所有元素的绝对值中**最小**的；即*X*的负无穷范数为：2；

Matlab代码：*X*=\[2, 3, -5, -7\]; XLfsf=norm(*X*,-inf);

## **2、矩阵的范数** ##

设：向量![X \\in \{R^n\}][X _in _R_n]，矩阵![A \\in \{R^\{n \\times n\}\}][A _in _R_n _times n]，例如矩阵*A*为：

*A*=\[2, 3, -5, -7;

4, 6,  8, -4;

6, -11, -3, 16\];

**（1）矩阵的1-范数（列模）：**![\{\\left\\| A \\right\\|\_1\} = \\mathop \{\\max \}\\limits\_\{X \\ne 0\} \\frac\{\{\{\{\\left\\| \{AX\} \\right\\|\}\_1\}\}\}\{\{\{\{\\left\\| X \\right\\|\}\_1\}\}\} = \\mathop \{\\max \}\\limits\_\{1 \\le j \\le n\} \\sum\\limits\_\{i = 1\}^n \{\\left| \{\{a\_\{ij\}\}\} \\right|\}][left_ A _right_1_ _ _mathop _max _limits_X _ne 0_ _frac_left_ _AX_ _right_1_left_ X _right_1_ _ _mathop _max _limits_1 _le j _le n_ _sum_limits_i _ 1_n _left_ _a_ij_ _right]；矩阵的每一列上的元素绝对值先求和，再从中取个最大的，（列和最大）；即矩阵*A*的1-范数为：27

Matlab代码：fs1=norm(A,1);

**（2）矩阵的2-范数（谱模）：**![\{\\left\\| A \\right\\|\_2\} = \\mathop \{\\max \}\\limits\_\{X \\ne 0\} \\frac\{\{\{\{\\left\\| \{AX\} \\right\\|\}\_2\}\}\}\{\{\{\{\\left\\| X \\right\\|\}\_2\}\}\} = \\sqrt \{\{\\lambda \_\{\\max \}\}(\{A^T\}A)\} = \\sqrt \{\\mathop \{\\max \}\\limits\_\{1 \\le i \\le n\} \\left| \{\{\\lambda \_i\}\} \\right|\}][left_ A _right_2_ _ _mathop _max _limits_X _ne 0_ _frac_left_ _AX_ _right_2_left_ X _right_2_ _ _sqrt _lambda _max _A_T_A_ _ _sqrt _mathop _max _limits_1 _le i _le n_ _left_ _lambda _i_ _right]，其中   ![\{\\lambda \_i\}][lambda _i]为![\{A^T\}A][A_T_A]的特征值；矩阵![70][]的最大特征值开平方根。

Matlab代码：fs2=norm(A,2);

**（3）矩阵的无穷范数（行模）：**![\{\\left\\| A \\right\\|\_\\infty \} = \\mathop \{\\max \}\\limits\_\{X \\ne 0\} \\frac\{\{\{\{\\left\\| \{AX\} \\right\\|\}\_\\infty \}\}\}\{\{\{\{\\left\\| X \\right\\|\}\_\\infty \}\}\} = \\mathop \{\\max \}\\limits\_\{1 \\le \{\\rm\{i\}\} \\le n\} \\sum\\limits\_\{j = 1\}^n \{\\left| \{\{a\_\{ij\}\}\} \\right|\}][left_ A _right_infty _ _ _mathop _max _limits_X _ne 0_ _frac_left_ _AX_ _right_infty _left_ X _right_infty _ _ _mathop _max _limits_1 _le _rm_i_ _le n_ _sum_limits_j _ 1_n _left_ _a_ij_ _right]；矩阵的每一行上的元素绝对值先求和，再从中取个最大的，（行和最大）

Matlab代码：fswq=norm(A,inf);

[https_blog.csdn.net_zaishuiyifangxym_article_details_81673491]: https://blog.csdn.net/zaishuiyifangxym/article/details/81673491
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