The tensor product, demystified (2018)
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The tensor product, demystified (2018)
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Re: The tensor product, demystified (2018)
#2Very well written. The author should write textbooks.
Re: The tensor product, demystified (2018)
#3A nice, high school level writeup of how to calculate this product with ordinary vectors, but it totally drops the ball on the necessity and its use in physics. It would probably be best to ignore the entire last paragraph and instead read up on pure vs. mixed quantum states if you actually care about that.
Re: The tensor product, demystified (2018)
#4Tensor product is basically the mathematical way to express if condition.
Say you have 2x2 matrix A, B, C.
Any arbitrary component within the Tensor(A, B, C) is:
if A =(a1,a2) and B=(b1,b2) and C=(c1,c2): this value
While in many computational natural science, people using tensor product to store and manipulate data. Also it is how the mathematical equations being written on paper.
But in computer science's perspective, dealing with tensor matrix is simply a waste of memory since 90% of the time people are dealing with sparse system. System that their matrix is dominated by zero. Also it would be super clear if people just write if-then pseudocode instead of cryptic half-bake tensor expressions. People tend to invent their own notation while writting paper.
Re: The tensor product, demystified (2018)
#5If only those squiggles were readable.
Re: The tensor product, demystified (2018)
#6It was a bit annoying that the first few examples give a flattened tensor product. It's explained at the end, but the dimensions are at least as important as the values and it feels wrong to skip that (isomorphism notwithstanding).
Re: The tensor product, demystified (2018)
#7Hmm, I always disliked coordinate approach to (multi)linear algebra. To me, the notion of a tensor product clicked with the following definition:
Tensor product v₁⊗v₂ of vectors form V is bilinear form on the dual space V* defined by v₁⊗v₂(ω₁,ω₂) = v₁(ω₁)v₂(ω₂). Tensor product V⊗V is space of all bilinear forms on V*,
This can easily be generalized to products of forms (covariant tensors), and mixed tensor products. Getting coordinate representation in basis (E₁ ... Eₙ) is trivial: (v₁⊗v₂)ₖ,ₗ = v₁⊗v₂(εₖ,εₗ) where εₖ is dual to Eₖ...
Re: The tensor product, demystified (2018)
#8Tensor product is basically the mathematical way to express if condition. Say you have 2x2 matrix A, B, C. Any arbitrary component within the Tensor(A, B, C) is: if A =(a1,a2) and B=(b1,b2) and C=(c1,c2): this value The coordination is concatenation of matching dimension: (a1b1c1, a2b2c2) While in many computational natural science, people using tensor product to store and manipulate data. Also it is how the mathemat…
> Tensor product is basically the mathematical way to express if condition.
Not really, no. The way mathematicians actually express if conditions is with the word "if". The obvious pointlessly formal way to do it is with a pair of functions `ThingConditionedOn -> {0, 1}` and `{0, 1} -> Result`, but why would you?
> dealing with tensor matrix is simply a waste of memory since 90% of the time people are dealing with sparse system.
Tensors are not their components, any more than locations are their coordinates. Whether you choose a sparse or dense (or symbolic) encoding does not change the object being encoded.
Re: The tensor product, demystified (2018)
#9Hmm, I always disliked coordinate approach to (multi)linear algebra. To me, the notion of a tensor product clicked with the following definition: Tensor product v₁⊗v₂ of vectors form V is bilinear form on the dual space V* defined by v₁⊗v₂(ω₁,ω₂) = v₁(ω₁)v₂(ω₂). Tensor product V⊗V is space of all bilinear forms on V*, This can easily be generalized to products of forms (covariant tensors), and mixed tensor products.…
I always dislike the unnecessary use of dual of dual. We should define tensor product as a quotient space of all linear combinations of v₁⊗v₂ as in https://en.wikipedia.org/wiki/Tensor_product#As_a_quotient_s...
The advantage of not using dual of dual is that it generalizes correctly to infinite dimensional vector spaces and modules.
Re: The tensor product, demystified (2018)
#10This didn't demystify anything to me, I left it more mystified it than when I entered.