[tagrun]

Which mathematician disagrees with what exactly?

Tensors are introduced by physicists to ensure various physical quantities (which involve coordinates and their derivatives) do not depend on the arbitrarily chosen coordinate system. This is ensured through the transformation properties of tensors.

The name tensor itself comes from the theory of elasticity, Cauchy stress tensor, which BTW is uniform in many practical cases, and obeys the following tensor transformation rule:

https://en.wikipedia.org/wiki/Cauchy_stress_tensor#Transform...

like any other (contravariant) tensor must.

Matrices are not examples of tensors. Matrices can be used for representation of tensors, in which case tensor product becomes Kronecker product, but matrices in general don't have to represent tensors. You can put anything, including your favorite colors or a list of random numbers, in a matrix, and it won't be a tensor in general, not unless it must transform like a tensor under coordinate system changes.

Similarly, TensorFlow "tensor" is just a multidimensional data array, with no transformation rules enforced on it, and therefore is not a tensor.

[clairity]

i did a whole phd-level course (a long time ago) in deformable materials, which was entirely based on tensors, and i still don’t know how to differentiate one from vectors/matrices. even the idea that tensors must obey coordinate transforms doesn’t really do it, since the practical applications of vectors/matrices do so as well.

it’s like some people invented a new word and won’t tell you what it actually means in sufficient detail to differentiate it from all the other words you know. so you keep using it with others in the hopes that contextual information will finally make it clear. one day.

[abdullahkhalids]

I just taught a module on Tensors in one of my physics courses. The mistake lots of people make is not show examples of matrices that are not tensors. But this is really difficult to do in physics courses because all physics matrix-like-objects must be tensors. Any theory that has non-tensor like objects in it will necessarily fail as soon as you change your coordinate system.

Thankfully, there is a great historical example of this. The electric field vector \vec{E} = (E_x, E_y, E_z), is not a tensor. It doesn't obey the tensor-transformation law. Similarly, the magnetic field vector is not a tensor. These are matrices, but not tensors.

As you know the Electromagnetic tensor [1] is the tensor that correctly transforms under coordinate transformation, and hence allows different observers to agree with each other.

[1] https://en.wikipedia.org/wiki/Electromagnetic_tensor

[clairity]

thanks for the examples. i vaguely recognize the term 'electromagnetic tensor', but i have to admit i didn't know that it was special in that way. i can barely spell 'tensor' at this point.

ps - one thing that always annoyed me was the limitation of linearity in so many of these models (which i totally understand why, but still). all the interesting real-world stuff happens non-linearly...

[abdullahkhalids]

Indeed it is annoying. But the models are linear not because physicists mistakenly believe that the world is linear, but because in most cases linear models are the only ones that one can solve to get qualitative predictions out. Non-linear models can be constructed as well, and then numerically solved on computers to get exact answers. But a physicist is one who understand the essential qualitative features of the world, rather than one who can compute understanding-free numerical answers.

In the words of Dirac, "I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it." This usually only works if your equations are linear.

[clairity]

yah, as an engineer[0], i totally get the solvability angle, and even the physicist's core desire to be able to test (and predict via) the math rather than the physical manifestations (which may be impossible to test directly), but i'm eager to see us advance deeper into the non-linear, since that's where it gets really interesting. like, how do proteins really work? or multi-body energy fields? we're in the infancy of really understanding all this stuff. the future is stochastic and non-linear. in a thousand years, people might look back with amusement on how ignorant we were with our puny little linear models and deterministic computers. =)

[0]: but at this point, not really. even in grad school, i only did linear modeling, and relatively rudimentary ones, at that.

[vcxy]

Interesting, you use the word matrix differently than me. The way I use it, a 2d array isn't necessarily a matrix. It's only a matrix if it represents a linear map between vector spaces with respect to chosen bases. Then again, I'm not much of a programmer, but I've taught linear algebra a few times. My head is just in a different place I guess.

[tagrun]

Any 2D array of data is a valid matrix.

But not any multidimensional data is a valid tensor.

[vcxy]

I'm not sure I buy this after you just used an example of a 2d array of you're favorite colors just a few minutes ago. Maybe I'm missing something. What kind of linear transformation does that represent, and between what vector spaces?

[xyzzyz]

(the exchange between you two is a perfect example of the disconnect between how mathematicians see linear algebra, and how physicists do)

[vcxy]

Haha, maybe so. I'm a mathematician, but based on some other comments I seem to be playing the role of physicist I think? Not sure.

[tagrun]

My favorite colors are 1, 2, 3 and 4 in some 8-bit grayscale color space. Here's a matrix

  [ 1 2 ]
  [ 3 4 ]

This represents some linear transformation, and is a matrix. Are you trying to argue otherwise?

FYI, matrices transform between vectors within a vector space.

[vcxy]

You have a list of labels you've conveniently decided to conflate with numbers. It's fine to use an enum for your data, but no, this data structure is not meant to convey a transformation.

You bring up a good point though, if this were meant to be a transformation, then we're talking about modules (Z/2^8Z being the underlying ring) and not vector spaces, which is fine. I was needlessly narrow when I said "vector spaces" earlier.

[tagrun]

You can have matrices of anything including rings, quaternions, octonions, dual numbers, matrices, vectors, bras, kets, etc etc, as long as you can multiply and add those objects. If you prefer real numbers, use the corresponding wavelength of the light, or voltage values in a photosensor, it doesn't matter. It still is a matrix.

Why do you mention things like data structures, enums and labels? This is math, not C++.

Anyway, the point is, any arbitrary 2D arrangement of numbers can be a valid matrix, whatever those numbers may represent.

[twangist]

> FYI, matrices transform between vectors within a vector space.

Square ones do, but m x n ones represent linear maps from an n-dimensional to m-dimensional vector space (over the/a field containing the elements of the matrix).