What is the Conceptual Difference Between a Matrix and a Tensor?

[dipole]

What is the conceptual difference between and matrix and a tensor? To me they seem like the same thing...

 

[lavinia]

What is the conceptual difference between and matrix and a tensor? To me they seem like the same thing...

why do you think that? explain.

 

[conquest]

Yeah I can see you're reasoning in fact you could actually express most tensors as a matrix in two certain bases. However a matrix can represent a linear function from 1 vector space to another and tensors specifically are multilinear functions on a certain cartesian product of a vector space and the duals of the same vector space. Thus they are linear on the tensor product and we can now express them on that in terms of a basis of this tensor product vector space.

Thus the key difference is that you can represent linear functions with matrices in a certain bases and there is a linear function implied by a tensor. This does not mean they are the same thing.

 

[D H]

What is the conceptual difference between and matrix and a tensor? To me they seem like the same thing...

First off, it depends on what you mean by matrices. If you mean some NxM construct, then the answer is that they are very different. There are zeroth order tensors, which are a special kind of scalar, first order tensors, which can be represented as vectors. Third order tensors can be presented as a NxNxN "matrix".

While tensors can be represented in the form of a matrix, that does not mean that they are matrices, and it most certainly does not mean that any old matrix is a tensor. Tensors are things that transform per a very strict set of rules. Just because you can denote the individual elements that form some aggregate by a set of indices does not mean that that aggregate is a tensor.

 

[AlephZero]

While tensors can be represented in the form of a matrix, that does not mean that they are matrices, and it most certainly does not mean that any old matrix is a tensor. Tensors are things that transform per a very strict set of rules. Just because you can denote the individual elements that form some aggregate by a set of indices does not mean that that aggregate is a tensor.

Some engineering representations of tensors defy the tensor transformation rules, to write something that looks like matrix algebra. A nice (?) example is the stress-strain relationship written in the "Voigt notation". http://en.wikipedia.org/wiki/Hooke's_Law#Anisotropic_materials

 

[dipole]

Well in physics we just write tensors like we would a matrix, and multiplying them by vectors is the same operation as matrix multiplication.

I guess you're saying they have different mathematical properties, but maybe I need some more linear algebra to understand.

 

[HallsofIvy]

A matrix can be used to represent a tensor, in a given basis (coordinate system), just as a vector can be represented by, say, <a, b, c>. Change the coordinate system and the matrix will change for the same tensor.

 

[conquest]

Mathematically you can best view a matrix as simply a way of putting things into rows and columns neatly. When you have a neat ordering you can then represent lots of things with this ordering like certain tensors, but also for instance graphs
http://en.wikipedia.org/wiki/Adjacency_matrix

So the mathematical object you refer to is a tensor the technique you use to compute or prove things using tensors may involve matrices (although sometimes this is not needed or even onnly makes things harder).

 

[dipole]

Ok, so is the term "matrix" even something which is a mathematically well-defined object then? Or are most things we call matrices really vectors?

For example, in optics you can make use of Ray transfer matrices:

http://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis

What specifically separates these from a tensor, vector, or something else called a matrix?

 

[wisvuze]

Matrices can represent k-linear maps. Any k-linear map can be expressed as a linear combination of tensors. ( for any sticklers out there, everything said here is in finite dimensions )

 

[wisvuze]

Ok, so is the term "matrix" even something which is a mathematically well-defined object then? Or are most things we call matrices really vectors?

A vector is just simply something that lives in a vector space ( an object of a space that satisfies a bunch of axioms ). A matrix is literally just an array of elements ( or numbers if you want ). The connection is formed when

1) bases are given for vector spaces
2) operations are defined for these rectangular arrays of numbers. Specifically,
linear operations +, - , scalar multiplication.. and a matrix multiplication operation that


when the above two are satisfied, a matrix can represent systems of any finite dimensional vector space , by assigning to its entries, a vector's "coordinates" with respect to a fixed basis. For example, with a basis {e1 , e2 , e3}, a vector
v = a1e1 + a2e2 + a3e3 can be represented as [ a1 a2 a3 ]. The matrix and the vector are still different objects.
Matrix multiplication is defined so that it corresponds with compositions of linear maps