Difference Between Matrices & Tensors: Explained

In summary, a matrix that satisfies certain tranform rules can be thought of as representing a tensor of rank 2. Tensors can have rank 0,1,2,3,4..., whereas matrix's are only representations and can have any rank. Tensors are also less general in a sense as they are restricted by transform laws.
  • #1
wintercarver
8
0
could someone please explain the difference or non-difference of matrices and tensors? i come across the two plenty in various fields of physics and am curious. i have a feeling this question has been asked and answered before, but i could not find a previous thread, so pointing me to another post would also be appreciated. thanks.

-wc
 
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  • #2
wintercarver said:
could someone please explain the difference or non-difference of matrices and tensors? i come across the two plenty in various fields of physics and am curious. i have a feeling this question has been asked and answered before, but i could not find a previous thread, so pointing me to another post would also be appreciated. thanks.

-wc
A matrix that satisfys certain tranform rules can be thought of as representing a tensor of rank 2. Tensors can have rank 0,1,2,3,4...
Thus scalars and vectors are tensors of ranks 0 and 1 respectively. Thus tensors are in a sense more general than matrix's as they are not representations, and they can have any rank. They are also less general in a sense as they are restricted by transform laws.
 
  • #3
the word tensor has two meanings, it is both a verb and a noun. it is a sort of product that can be performed between two vector spaces, and then it is an element of such a product.

when you tensor multiply a vector space V by the dual W* of another vector space W, the result maps naturally to the space of linear transformations from W to V.

when the spaces are finite dimensional, this map is an isomorphism. one also knows that this space of linear transformations is isomorphic, non naturally, to the space of matrices of size dim(V) by dim(W).

i myself am not too up on this rank language, but my impression from reading other people posts is that when the space V is always the same, say R^n, then one multiplies together exclusively copies of V and V*, and the rank refers to the number of copies of each one.

thus i would have thought such users would have called a tensor with one copy of each, i.e. a matrix, a "rank (1,1) tensor", whereas rank 2 would have meant a product of two copies of V.

of course for people who routinely choose bases, the distinction between V and V* is much less clear, hence one cannot distinguish between rank(1,1) and rank 2.

but that is not my area.
 
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  • #4
I'm sure I've heard a "rank (m, n) tensor" also called a "rank m+n tensor" before.
 
  • #5
yes no doubt that explains it.
 
  • #6
A tensor of rank (q,r) on a vector space of dim(n) over a field Q forms a vector space of dim(n^(q+r)), so the set tensors of rank (1,1), (2,0) and (0,2) all form vector spaces of dim(n^2), simlairly the set of nxn matrices over a field Q form a vector space of dim(n^2), so in terms of structure as vector spaces (i.e. when addition, subtraction and scalar multiplication is concerned) there is no difference between a rank 2 tensor and an nxn matrix. Both tensors and matrices have more properties (that much should be obvious from the fact that we choose to distnguish rank (1,1) and rank (2,0), etc tensors) than just those associated with the basic properties vector spaces they form, so a rank 2 tensor is not the same as an nxn matrix (though they do share other properties which do extend the usefulness of representing rank 2 tensors as matrices beyond addition, subtraction and scalar multiplication).
 
  • #7
the difference is clear mathematically, but subtle if you are using them in an applied way.

consider the metric tensor for example:

[tex]G = g_{ij} dx^i \otimes dx^j[/tex]

some people call [itex]g_{ij}[/tex] (the elements of a matrix) the metric tensor, but this is not mathematically correct, they form the components of the metric tensor. this is analogous to calling the components of a vector the vector itself.

the reason for this is since

[tex]dx^i(\vec{v}) = v^i[/tex]

then

[tex]G(\vec{u}, \vec{v}) = u^i g_{ij} v^j[/tex]

which is nothing more than

[tex]\begin{pmatrix}u^1 & u^2 & ...\end{pmatrix}\begin{pmatrix}g_{11} & g_{12} & ... \\ g_{21} & g_{22} & ... \\ ... & ... & ...\end{pmatrix}\begin{pmatrix}v^1 \\ v^2 \\ ...\end{pmatrix}[/tex]
 
  • #8
now through the magic of televisison, mr science will demonstrate how to use the dot product to change a 2 tensor into a 1,1, tensor! look children, vtensw appears to be a 2 tensor, but behold, if t is a vector then we can wave our dot product wand and say: presto: you are a 1,1 tensor and now when vtensw sees the vector t,

it pounces and yields the vector <v,t>w !@.

observant boys and girls can no doubt guess how to change it as well into a 2 tensor of opposite variance, yielding a number, when confronting a pair of vectors, (s,t) ??

mirabile dictu.

the moral is, one recognizes a wolf only after it tries to eat the sheep.

i.e. one often does not know what object v(tens)w is until the great oz says so: i.e. if you say it is a (0,2) tensor then it is, and if you say it is a (1,1) tensor then it is that as well. enjoy the game! (and always be a student of behavior.)
 
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  • #9
But always wear a helmet while playing this game!
 
  • #10
some of us apparently forgot to. as lyndon johnson used to say. :wink:
 
  • #11
Isn't WC simply asking for the difference between a matrix and a tensor? The answer to that is that a matrix is only one way of *representing* a tensor. It could alternatively be represented by a single letter, bold-faced, underlined or whatever. Matrices can be used for other purposes, e.g to represent non-tensorial objects such as the coefficients of a bank of simultaneous equations.
 

1. What is the main difference between matrices and tensors?

Matrices and tensors are both mathematical objects used to represent data in a structured way. However, the main difference between the two is their number of dimensions. Matrices are two-dimensional arrays, while tensors can have any number of dimensions, including higher dimensions such as 3, 4, or even more.

2. Can matrices and tensors be used interchangeably?

No, matrices and tensors cannot be used interchangeably. They have different properties and operations associated with them. For example, matrices have well-defined addition and multiplication operations, while tensors have more complex operations such as contraction and outer product.

3. How are matrices and tensors related mathematically?

Matrices can be seen as a special case of tensors, where the number of dimensions is limited to two. This means that all operations and properties of matrices can be applied to tensors, but not vice versa. Additionally, tensors can be used to represent matrices in higher dimensions.

4. What are some common applications of matrices and tensors in science and engineering?

Matrices and tensors have a wide range of applications in various fields such as physics, engineering, and computer science. Some common applications include data analysis and machine learning, image and signal processing, quantum mechanics, and fluid dynamics.

5. Are there any limitations to using tensors compared to matrices?

Tensors have a more complex structure compared to matrices, which can make them more difficult to work with. Additionally, computations involving tensors can be more computationally expensive. However, tensors are more versatile and can represent more complex data structures, making them necessary for certain applications.

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