# It is correct to say that a tensor is simply a multidimensional array of related quantities? But what about a tensor as a transformation?

It is correct to say that a tensor is simply a multidimensional array of related quantities?

More specifically a tensor is a collection or tuples of vectors where every vector in the tuple represent a different type of information but the components of the different vectors depend of each other.

I said this because of the following sentence I read in Wikipedia:

"at the end of the 19th century, the electromagnetic field was understood as a collection of two vector fields in space. nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime"

I understand that electric and magnetic fields are grouped into a tensor because the components the magnetic field depend of some maner of the components of the electric field and viceversa.

But what about tensors as transformation objects? if you have two vectors (not tuples of vectors) then the transformation is simply a matrix (or rank 2 tensor), but what is the necessity for tensors of rank bigger than 2?

I'm asking several things: i) It is correct to say that a tensor as a quantity is a tuple of vectors, that is needed to group vectors of different types that are related like the magnetic and electric fields ii) Can a tensor by viewed as linear transformation? In this sense how a tensor is different from a regular matrix? It is because it transform tuples of vectors instead of simple vectors? iii) What is an example of a tensor as a linear transformation in physics?

• Could you clarify, please? Some tensors represent physical quantities; some represent transformations; some represent both. Is that what you’re asking? Or is this somehow about tensor of high rank? Jul 4, 2018 at 15:47
• Hi Bob, I'm asking several things: i) It is correct to say that a tensor as a quantity is a tuple of vectors, in contrast with a normal vector? ii) Can a tensor by viewed as linear transformation? In this sense how a tensor is different from a regular matrix? It is because it transform tuples of vectors instead of simple vectors? iii) What is an example of a tensor as a linear transformation in physics? Jul 4, 2018 at 15:55
• Thank you. Could you edit those into your question, please? Jul 4, 2018 at 15:57
• Done, I have added those to my question. Jul 4, 2018 at 16:09
• Related: physics.stackexchange.com/q/20437/2451 and links therein. Jul 4, 2018 at 19:17

A tensor is a multidimensional array of related quantities which has specific behaviour under transformations of co-ordinates.

If $a$ and $b$ are vectors then the Cartesian product $T=a \times b$ i.e. $T_{ij}=a_ib_j$ is a tensor, and the transformation of $T$ is specified by the transformations if $a$ and $b$. This can be generalised to tensors of any order, and to upper/lower indices to handle contravariant/covariant vectors $T^i{}_j{}^k{}_\ell = a^ib_jc^kd_\ell$ etc.

So any (reasonable) cartesian product gives a tensor - though not all tensors can be written as cartesian products. However any tensor has to have the same transformation properties as the equivalent cartesian product. Iff a matrix has this property then it is a tensor

TLDR: Tensor mathematics is a tool. Like many good tools, it can be used in multiple ways.

It's better to think about the properties of the tool and of its uses as separate things: People do use screwdrivers as little pry bars, but it's not particularly useful to discuss "Is a screwdriver a pry bar?"

In physics, the general aspect of tensors (including their limiting cases, vectors and scalars) is that they represent something that's coordinate-independent (which is necessary for them to be "real"). That means that they transform in a specific way when moving from one coordinate system to another.

i) It is correct to say that a tensor as a quantity is a tuple of vectors, that is needed to group vectors of different types that are related like the magnetic and electric fields

Some are. Not all are. In physics, what matters to whether something is a tensor is how something transforms, not whether it transforms something else.

Transformations are easiest to see in the tuple of vectors. A matrix that transforms a vector from one basis (i.e. x, y, z) to another basis (x', y', z') consists of the set of vectors expressing the x direction in x', y, z', the y direction, and the z direction. The transform is just finding that combination of those new basis vectors that represent the x,y,z components of the "true" (non-component, real thing) vector.

But not all tensors have that property. What matters is what the tensor represents. For example, it's not particularly useful to think of the mass-energy tensor of general relativity as a transform.

I'm not sure it's useful to think of the $A^{\mu \nu}$ tensor of electrodynamics as "group(ing) vectors of different types that are related like the magnetic and electric fields". It's clearly not a "tuple of vectors".

ii) Can a tensor by viewed as linear transformation? In this sense how a tensor is different from a regular matrix? It is because it transform tuples of vectors instead of simple vectors?

"viewed as"? Not sure that's a useful way to think about physics. Tensors are tools: some represent transforms, but others don't.

A matrix is a particular kind of tensor. So is a vector, for that matter.

But not all matrices represent coordinate-independent quantities. Financial data in an Excel spreadsheet is in some sense a matrix, but it's not coordinate-independent in any meaningful sense, and therefore not a tensor.

iii) What is an example of a tensor as a linear transformation in physics?

Rotations are the easiest example: A rotation matrix is a tensor. If you view that rotation from another coordinate system, the rotation is still physically the same, but the numbers in the rotation matrix will be different.

The Lorentz transform is another (though in some sense it's the same, as a Lorentz transform is a 4D rotation)