Is the Tensor Product of Two Matrices Really This Simple?

[space-time]

I was just watching a video that was reviewing some linear algebra, and it said that this was the tensor product:

Let's say you have a matrix A and a matrix B (both 2 by 2 matrices). If I want to calculate the tensor A ⊗ B, then the answer is basically just a matrix of matrices. In other words, I do this:

The first matrix of the tensor product space is: the scalar multiplication of A₁₁ * B
The 2nd matrix of the tensor product space is: A₁₂ * B
The 3rd matrix is: A₂₁ * B
The last matrix is: A₂₂ * B

Over all, this makes a 4 by 4 matrix (which I will call C even though I know it should really be denoted A ⊗ B) with elements:

C₁₁ = A₁₁ * B₁₁
C₁₂ = A₁₁ * B₁₂
C₁₃ = A₁₂ * B₁₁
C₁₄ = A₁₂ * B₁₂
C₂₁ = A₁₁ * B₂₁
C₂₂ = A₁₁ * B₂₂
C₂₃ = A₁₂ * B₂₁
C₂₄ = A₁₂ * B₂₂

C₃₁ = A₂₁ * B₁₁
C₃₂ = A₂₁ * B₁₂
C₃₃ = A₂₂ * B₁₁
C₃₄ = A₂₂ * B₁₂
C₄₁ = A₂₁ * B₂₁
C₄₂ = A₂₁ * B₂₂
C₄₃ = A₂₂ * B₂₁
C₄₄ = A₂₂ * B₂₂

I just want to ask: Is this really all there is to taking a tensor product? Is this really the process or is this just some simplified special case? I just ask this because I have asked on threads before about tensor products and tried to look up videos and web pages on them, and every time my source has just made it out to be some daunting process that was so difficult to explain and just about impossible to show an example of.

 

[fresh_42]

In coordinate form, your example is correct. An example of the tensor product of two matrices. In general there may be also sums of them to get other elements in the vector space of here $\mathbb{M}_{2 \times 2} \otimes \mathbb{M}_{2 \times 2}$.

Edit: One can arrange them in different ways, e.g. as four layers of $2 \times 2$ matrices: $A_{11}B ,...$