norm：翻译为模或者内积，广义来说是一个函数

vector（向量） norms

1. eculidean（欧几里得）norm

vector (x = (x_1;x_2; …; x_n))
其eculidean norm为 ：(||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 imes 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}
]

坚持