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.
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.
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.
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...
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.
[0]: but at this point, not really. even in grad school, i only did linear modeling, and relatively rudimentary ones, at that.
But not any multidimensional data is a valid tensor.
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FYI, matrices transform between vectors within a vector space.
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.
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.
I'm genuinely claiming they aren't numbers due to the explicit context. The entries are colors with numerals as labels.
Now, you've said something slightly different where I think we can agree. If the entries are from a set closed under a notion of multiplication and addition, then it represents a matrix even if it isn't being used in that way. But I still say that if you're merely using the array as a place to keep some data then I won't be using the word matrix, I'll just call it an array.
Anyway. I started this by saying it was interesting that we use the word matrix slightly differently. I still think that's interesting and think it's totally fine if you want to call an array a matrix even if the entries don't come from something where a transformation makes sense. I wouldn't "correct" someone's usage. I just think it's interesting how we use it different. Anyway, I feel like you just think I'm dumb, so I'm ending this here. You might not, but it's hard to read the vibe via text. If you have something to add you think will get me to change my word choices, feel free to respond and I'll read it, but I'm not replying on this thread anymore.
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).
Sure there are: Any basis of the underlying vector space(s) induces a basis of the tensor space. Components respective to some basis are coordinates. You can then investigate what happens to the induced basis (or rather, the respective components) under a basis transformation of the underlying vector space(s), which is where the "physicist's" definition of tensors originates.
A change of coordinates does indeed induce a change of basis, but a change of basis isn't really a change of coordinates. And strictly speaking some vector spaces don't really have an obvious basis (without invoking choice), so having a basis be a prerequisite for the definition is not ideal.
The whole requirement that a tensor is 'something that transforms like [...] under a coordinate transformation' is just how physicists have chosen to phrase that a vector bundle is only well defined if it's definition isn't dependent on some arbitrary choice of coordinates. In my opinion this requirement is more easily apparent in the mathematical definition where there is no choice of coordinates in the first place, rather than the physicists way of working with some choice of coordinates and checking how things transform.
However, physicists get introduced to tensors far earlier than any excursions into differential geometry when discussing rigid bodies.
Matrices (or linear transformations in general) are important examples of tensors. There's a nice adjunction between tensor spaces A(x)B and the space of linear transformations B=>C given by:
Hom(A(x)B, C) = Hom(A, B=>C)
In the case of Tensorflow I think they do actually still talk about linear transformations of some kind so it's perfectly fine to call them tensors.