Tensors for Beginners 13: Tensor Product vs Kronecker Product

[zhiiology]

Hi eigenchris, just wanted to say that while the video is clear, the kronecker product is usually computed by shoving the tensor on the right into the tensor on the left, which is usually how it's done at least in quantum computing

[zzzoldik]

tankyou, eigenchris, i never understand if i learn tensor before, you explain it from the star / beginner. these really helpful

[anthonym]

My man this is exactly what I needed thank you

[vassillenchizhov]

Your notation with the array of arrays suggests that the Kronecker product produces arrays of higher dimension that two. However the Kronecker product of a matrix is always a matrix, for instance if I take $u\oplus v \oplus w$ this will not produce an object having three indices, it will still be a matrix. Of course you can relate this to the object made from arrays of arrays, but technically they are not the same thing.

[alessandrogardini]

Hi! It seems to me that at 2:49 the new vector components should be represented by a latin letter and upper indices (like [w^1 over w^2] rather than [omega_1 over omega_2]). Thank you again for the videos.

[underratedPie]

I told my colleague to study the Tensor calculus on this channel only

[mathgeek]

Something seems off at 1:36. It is my understanding that the tensor product takes a (p, q)-tensor and (a, b)-tensor and produces a (p+a, q+b) tensor. In the example shown, we have the vector e_i which is a (1, 0) tensor and eps^j which is a (0, 1) tensor. This means that the tensor product between e_i and eps^j is a (1, 1) tensor. What's confusing is about 1:36 is that the tensor resulting from the tensor product of e_i and eps^j should have a covector and a vector as an input. This will make sure that the output of the tensor is a scalar and not a vector.

[jexwtxk]

If we can download all the slides, it would be perfect, since we can print it out and study it without electronic devices, also we can drop down notes for deeper understandings.

[no-one-in-particular]

2:07 The "circle times" shouldn't be there from the second row onwards as epsilon acting on v is a number

[geneyi]

I didn't get it that what is the definition of Kronecker product? "We just distribute the array on the left into the array on the right."

[jacobshin]

Would the array at 3:02 be a rank 3 tensor?

Am I correct in guessing that the rank = m+n, where m is the # of covariants and n is the number of contravariants?

[jasonbroadway]

I am hazy now. I followed you until this video and the previous video.

[Timelaser]

Hi, I'd just like to clarify something. On 2.21, is the summation sign implied for v(j)e(i)? I understand Einstein's notation is used for letters which are the same on top and the bottom, but in this case the letters at the top and bottom are different.

[auvski]

I'm actually pretty sure the Kronecker product as used in this video (and the previous one) is backwards. Both in my university's coursework and on the Wikipedia page, it seems like you're actually supposed to distribute the array on the right over the one on the left, that is, the left array is meant to be used as a "template" and the right array is copied for each block. You can take a look here:

[jamescook]

Hold a second... now a row vector is an array. Good.

[thedorantor]

1:57 so I can just as well put a co-vector in as argument, the epsilon of the co-vector "eats" the e_i of the (1, 1) tensor, and the result would be a co-vector?

[trendytrenessh]

Same thing, different context
Got it, cheers

[PaulWintz]

At 1:56, when the j-th basis covector of V* acts on the k-th basis vector of V to become the Kronecker delta, why does the tensor product operator disappear? Is it a valid operation to take the tensor product between a vector and a scaler? And does it merely equal the scalar times the vector?

[dirrelito]

Typically, you also flatten when using the Kronecker product. Tensor product increases tensor order, but kronecker product does not. This is a quite imporant distinction both in theory and practise. The Kronecker product as you explain it here is not how it works in e.g. Numpy or PyTorch.

[minecrafthowtodude]

so.... I understood a little bit of that :)