norm：翻譯為模或者內積，廣義來說是一個函數
vector（向量） norms
1. eculidean（歐幾里得）norm
vector \(x = (x_1;x_2; ...; x_n)\)
其eculidean norm為 ：\(||x|| = \sqrt{x^T x} = (\sum_{i=1}^n x_i^2)^{\frac 12} = \sqrt{x_1^2 +x_2^2 + ...+x_n^2}\)
2. P norm (P>=1)
\[||x||_p = (\sum_{i=1}^n {|x_i|}^p)^{\frac 1p} \]
常用：1 norm 、2norm（eculidean norm）、\(\infty\) norm
matrix norms
there have a matrix A \(\in C^{m \times n}\)
1. frobenius matrix norm (F norm)
\[||x||_F^2 = \sum_{ij}{|a_{ij}|}^2 = \sum_i {||A_{i*}||}_2^2 =\sum_j {||A_{*j}||}_2^2 = trace(A^* A) \]
2. matrix 2-norm
\[||A||_2 = \sqrt {\lambda_{max}} \]
\[\lambda_{max} \quad {是A^* A 最大特征值} \]
3. matrix 1-norm
\[||A||_1 = \, {列向量最大1模} \]
4. matrix \(\infty\) norm
\[||A||_{\infty} =\, {行向量最大1模} \]