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In mathematics, theHadamard product(also known as theelement-wise product,entrywise productorSchur product) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.
在数学中，阿达玛乘积 (Hadamard product，又译哈达玛乘积)，又名舒尔乘积 (Schur product) 或逐项乘积 (entrywise product)，是一个二元运算，其输入为两个相同形状的矩阵，输出是具有同样形状的、各个位置的元素等于两个输入矩阵相同位置元素的乘积的矩阵。

entry ['entri] n. 记录；词条；登录；录入

The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions.

1. Definition

For two matricesA \mathbf {A}AandB \mathbf {B}Bof the same dimensionm × n m \times nm×n, the Hadamard productA ⊙ B \mathbf {A} \odot \mathbf {B}A⊙B(sometimesA ∘ B \mathbf {A} \circ \mathbf {B}A∘B) is a matrix of the same dimension as the operands, with elements given by

( A ⊙ B ) i j = ( A ) i j ( B ) i j . (\mathbf {A} \odot \mathbf {B})_{ij} = (\mathbf {A})_{ij} (\mathbf {B})_{ij}.(A⊙B)ij​=(A)ij​(B)ij​.

For matrices of different dimensions (m × n m \times nm×nandp × q p \times qp×q, wherem ≠ p m \neq pm=porn ≠ q n \neq qn=q), the Hadamard product is undefined.

3 × 3 3\times 33×3矩阵A \mathbf {A}A与3 × 3 3\times 33×3矩阵B \mathbf {B}B的阿达玛乘积为：

[ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] ∘ [ b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ] = [ 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 ] . \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}.​a11​a21​a31​​a12​a22​a32​​a13​a23​a33​​​∘​b11​b21​b31​​b12​b22​b32​​b13​b23​b33​​​=​a11​b11​a21​b21​a31​b31​​a12​b12​a22​b22​a32​b32​​a13​b13​a23​b23​a33​b33​​​.

2. Properties

• The Hadamard product is commutative (when working with a commutative ring), associative, and distributive over addition.
  阿达玛乘积满足交换律 (当使用交换环时), 结合律和对加法的分配律

That is, ifA \mathbf {A}A,B \mathbf {B}B, andC \mathbf {C}Care matrices of the same size, andk kkis a scalar:

A ⊙ B = B ⊙ A A ⊙ ( B ⊙ C ) = ( A ⊙ B ) ⊙ C A ⊙ ( B + C ) = A ⊙ B + A ⊙ C ( k A ) ⊙ B = A ⊙ ( k B ) = k ( A ⊙ B ) A ⊙ 0 = 0 ⊙ A = 0 \begin{align} A \odot B &= B \odot A \\ A \odot (B \odot C) &= (A \odot B) \odot C \\ A \odot (B + C) &= A \odot B + A \odot C \\ (kA) \odot B &= A \odot (kB) = k(A \odot B) \\ A \odot 0 &= 0 \odot A = 0 \end{align}A⊙BA⊙(B⊙C)A⊙(B+C)(kA)⊙BA⊙0​=B⊙A=(A⊙B)⊙C=A⊙B+A⊙C=A⊙(kB)=k(A⊙B)=0⊙A=0​​

3. In programming languages

The NumPy numerical library interpretsa*bora.multiply(b)as the Hadamard product, and usesa@bora.matmul(b)for the matrix product.

References

[1] Yongqiang Cheng (程永强), https://yongqiang.blog.csdn.net/
[2] 动手学深度学习, https://zh.d2l.ai/index.html
[3] Deep Learning Tutorials, https://neuralthreads.medium.com/i-was-not-satisfied-by-any-deep-learning-tutorials-online-37c5e9f4bea1
[4] Gradient boosting performs gradient descent, https://explained.ai/gradient-boosting/descent.html
[5] Matrix calculus, https://en.wikipedia.org/wiki/Matrix_calculus
[6] Artificial Inteligence, https://leonardoaraujosantos.gitbook.io/artificial-inteligence
[7] Hadamard product, https://en.wikipedia.org/wiki/Hadamard_product_(matrices)