Tensor Product

Creator

Creator

Seonglae Cho

Created

Created

2024 Apr 19 12:53

Editor

Editor

Seonglae Cho

Edited

Edited

2024 Apr 23 5:11

Refs

Refs

Matrix Multiplication 과 다르다

Specialization of the Tensor Products

Kronecker product

A product like Id⊗𝑊 (with identity on the left) represents multiplying each position in our context by a matrix.

A product like 𝐴⊗Id (with identity on the right) represents multiplying across positions.

A product like 𝐴⊗𝑊 multiplies the vector at each position by 𝑊 and across positions with 𝐴. It doesn't matter which order you do this in.

The products obey the mixed-product property (𝐴⊗𝐵)⋅(𝐶⊗𝐷)=(𝐴𝐶)⊗(𝐵𝐷).

Hadamard product (
Jacques Hadamard)

동일한 차원 가질 때 pointwise

In mathematics, the tensor product V ⊗ W {\displaystyle V\otimes W} of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V × W → V ⊗ W {\displaystyle V\times W\rightarrow V\otimes W} that maps a pair ( v , w ) ,   v ∈ V , w ∈ W {\displaystyle (v,w),\ v\in V,w\in W} to an element of V ⊗ W {\displaystyle V\otimes W} denoted v ⊗ w {\displaystyle v\otimes w} .

https://en.wikipedia.org/wiki/Tensor_product

Kronecker product

In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product.[1]

Kronecker product

https://en.wikipedia.org/wiki/Kronecker_product

Hadamard product (matrices)

In mathematics, the Hadamard product is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements. This operation can be thought as a "naive matrix multiplication" and is different from the matrix product. It is attributed to, and named after, either French mathematician Jacques Hadamard or German mathematician Issai Schur.

Hadamard product (matrices)

https://en.wikipedia.org/wiki/Hadamard_product_(matrices)

Hadamard product (matrices)

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