# 阿達瑪乘積 (矩陣)

## 定义

${\displaystyle (A\circ B)_{ij}=(A)_{ij}(B)_{ij}.}$

## 样例

${\displaystyle 3\times 3}$ 矩阵A${\displaystyle 3\times 3}$ 矩阵B的哈达玛积为：

${\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}\circ {\begin{bmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{bmatrix}}={\begin{bmatrix}a_{11}\,b_{11}&a_{12}\,b_{12}&a_{13}\,b_{13}\\a_{21}\,b_{21}&a_{22}\,b_{22}&a_{23}\,b_{23}\\a_{31}\,b_{31}&a_{32}\,b_{32}&a_{33}\,b_{33}\end{bmatrix}}.}$

## 性质

{\displaystyle {\begin{aligned}&\mathbf {A} \circ \mathbf {B} =\mathbf {B} \circ \mathbf {A} ,\\&\mathbf {A} \circ (\mathbf {B} \circ \mathbf {C} )=(\mathbf {A} \circ \mathbf {B} )\circ \mathbf {C} ,\\&\mathbf {A} \circ (\mathbf {B} +\mathbf {C} )=\mathbf {A} \circ \mathbf {B} +\mathbf {A} \circ \mathbf {C} .\end{aligned}}}

## 参考资料

1. ^ Davis, Chandler. The norm of the Schur product operation. Numerische Mathematik. 1962, 4 (1): 343–44. doi:10.1007/bf01386329.
2. ^ Horn, Roger A.; Johnson, Charles R. Matrix analysis. Cambridge University Press. 2012.