Frobenius范数 迹运算

1. 设Am \times n的矩阵,其Frobenius范数定义为:

{\parallel A \parallel }_F = \sqrt{ \sum_{i=1}^{m} \sum_{j=1}^{n}  {\mid a_{ij} \mid }^2 }

2. 迹运算返回的是矩阵对角元素的和:

Tr(A) = \sum_{i} A_{i, i}

3. 迹运算提供了另一种描述矩阵Frobenius范数的方式:

{\parallel A \parallel }_F = \sqrt{ Tr(AA^T) }

证明:

 A = \begin{bmatrix} a_{11} & a_{12}  \\ a_{21} & a_{22} \end{bmatrix}             AA^T = \begin{bmatrix} a_{11} & a_{12}  \\ a_{21} & a_{22} \end{bmatrix}  \begin{bmatrix} a_{11} & a_{21}  \\ a_{12} & a_{22} \end{bmatrix} = \begin{bmatrix} a_{11}^2 + a_{12}^2 & ... ...  \\ ... ... & a_{21}^2 + a_{22}^2 \end{bmatrix}

{\parallel A \parallel }_F = \sqrt{a_{11}^2 + a_{12}^2 + a_{21}^2 + a_{22}^2 } = \sqrt{ Tr(AA^T) }

4. Tr(A) = Tr(A^T)

5. Tr(ABC) = Tr(CAB) = Tr(BCA)

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