Hadamard product {阿达玛乘积}
- [1. Definition](#1. Definition)
- [2. Properties](#2. Properties)
- [3. In programming languages](#3. In programming languages)
- References
In mathematics, the Hadamard product (also known as the element-wise product, entrywise product or Schur 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]
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The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions.
1. Definition
For two matrices A \mathbf {A} A and B \mathbf {B} B of the same dimension m × n m \times n m×n, the Hadamard product A ⊙ B \mathbf {A} \odot \mathbf {B} A⊙B (sometimes A ∘ 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 n m×n and p × q p \times q p×q, where m ≠ p m \neq p m=p or n ≠ q n \neq q n=q), the Hadamard product is undefined.
3 × 3 3\times 3 3×3 矩阵 A \mathbf {A} A 与 3 × 3 3\times 3 3×3 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}. a11a21a31a12a22a32a13a23a33 ∘ b11b21b31b12b22b32b13b23b33 = a11b11a21b21a31b31a12b12a22b22a32b32a13b13a23b23a33b33 .
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, if A \mathbf {A} A, B \mathbf {B} B, and C \mathbf {C} C are matrices of the same size, and k k k is 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 interprets a*b or a.multiply(b) as the Hadamard product, and uses a@b or a.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)