Thread: Scalar, Vector, Matrix, Tensor

1. I read that there's a sort of natural extension from scalars to vectors to matrices to tensors. The way I understand it, a scalar is a simple value. Extending on to a vector gives not only a value, but a direction. This is where I'm kind of stuck...how is a matrix an extension of the concept of a vector? Going on what I know, which isn't much, I'd say that perhaps it represents all the possible points a specific vector can point to, or something along those lines... And then tensors...does a tensor kind of represent all possible paths something can take in not only space but time as well? That's kind of what I've picked up... Any serious contributions are much appreciated.  2.

3. From what i understand:

A scalar has magnitude but no direction
A vector has magnitude and one direction
A tensor has magnitude and multiple directions

Tensors are represented by matrices, and can come in different 'orders' like a dyad or tryad, which are order 2 and 3 respectively (I think?). I think the matrices can be multi-dimensional as well, so that you could have a 4-dimensional matrix.  4. Also, tensors are apparently very hard/impossible to visualize... due to their multi-dimensional property I suppose. I also think that vectors can be thought of as matrices consisting of only one column. afaik linear algebra is the subject that deals with all of this, and you might be able to find some more formal definitions there.  5. I think of a scalar as being a value or a number. It's fine to think of it as being a magnitude, but note that this interpretation also requires you to keep track of the sign of the scalar if you're dealing with real numbers or the argument of the scalar if you're dealing with complex numbers. You can look at more general structures called fields and division algebras, and over these structures the magnitude interpretation doesn't necessarily make sense, so I tend to stick to number or value.

A vector, to me, is something that can be acted on by a scalar by multiplication and that can be added to another vector (in the same space). This is an abstract definition, but it's the most useful for arbitrary settings. In the finite dimensional case, though, I think of vectors as being lists of scalars. When you're dealing with real numbers, then the geometric picture makes sense--a vector corresponds to a magnitude and direction. Over the complex numbers, this breaks down a little bit--you need to work a little harder to say exactly what you mean by this. So I just go with the list of scalars interpretation.

A matrix relates vectors to each other. Matrices represent linear transformations, which are maps between two vector spaces that respect addition and scalar multiplication. I use this interpretation most often. For square matrices, the domain and range can be thought of as the same space, so I think of square matrices as representing motions--rotations, scalings, and shearings, for example.

But matrices can be thought of in a lot of other ways. Matrices are collections of vectors. Connecting this to the linear transformation interpretation, you can take a basis of the domain, look at its image under the transformation, and then think of the matrix as the list of image vectors you get. Matrices can also represent all sorts of other objects--products between two vectors spaces, such as the dot product, for example.

Tensors... I don't think I can talk about these without confusing the audience. My understanding of tensors is purely algebraic, and it's pretty abstract at that. I don't have much experience using tensors in geometry or physics, which is the interpretation you guys are probably going for, so I'm going to keep mute.  6. Originally Posted by serpicojr
Tensors... I don't think I can talk about these without confusing the audience. My understanding of tensors is purely algebraic, and it's pretty abstract at that. I don't have much experience using tensors in geometry or physics, which is the interpretation you guys are probably going for, so I'm going to keep mute.
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