Difference Between Tensor Product and Outer Product

MHB
Thread starter Sudharaka
Start date Mar 25, 2014
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Difference Outer product Product Tensor Tensor product

In summary, the tensor product and outer product are two mathematical operations used in linear algebra to combine two vectors or matrices. The main difference between them is the type of output they produce and the mathematical properties they follow. The output of a tensor product is a tensor, which is a multidimensional array, while the output of an outer product is a matrix. The tensor product is commutative, associative, and follows the distributive property, while the outer product is not commutative, not associative, and does not follow the distributive property.

Mar 25, 2014

Sudharaka

Gold Member

MHB

Hi everyone, :)

Xristos Lymperopoulos on Facebook writes (>>link<<);

can someone explain me the diffrence between tensor product of matrices and outer product of matrices?

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Sep 5, 2022

Wrichik Basu

Science Advisor

Insights Author

Gold Member

Mathematics SE has two questions on the same:

https://math.stackexchange.com/q/1951946/429032

https://math.stackexchange.com/q/1757901/429032

Related to Difference Between Tensor Product and Outer Product

1. What is the difference between tensor product and outer product?

The tensor product and outer product are both mathematical operations used in linear algebra to combine two vectors or matrices. The main difference between them is the type of output they produce and the mathematical properties that they follow.

2. What is the output of a tensor product?

The output of a tensor product is a tensor, which is a multidimensional array that represents the combined information of the two input vectors or matrices. The dimensions of the tensor are the product of the dimensions of the input vectors or matrices.

3. What is the output of an outer product?

The output of an outer product is a matrix, which is a two-dimensional array that represents the outer product of the two input vectors or matrices. The dimensions of the matrix are the product of the dimensions of the input vectors or matrices.

4. What are the mathematical properties of a tensor product?

The tensor product is commutative, meaning the order of the input vectors or matrices does not affect the output. It is also associative, meaning the grouping of the input vectors or matrices does not affect the output. Additionally, it follows the distributive property, meaning it distributes over addition and scalar multiplication.

5. What are the mathematical properties of an outer product?

The outer product is not commutative, meaning the order of the input vectors or matrices affects the output. It is also not associative, meaning the grouping of the input vectors or matrices affects the output. It does not follow the distributive property, as the outer product of two vectors cannot be expressed as the sum of the outer products of their individual components.