# Qualitatively, what is the difference between a matrix and a tensor?

Qualitatively (or mathematically "light"), could someone describe the difference between a matrix and a tensor? I have only seen them used in the context of an undergraduate, upper level classical mechanics course, and within that context, I never understood the need to distinguish between matrices and tensors. They seemed like identical mathematical entities to me.

Just as an aside, my math background is roughly the one of a typical undergraduate physics major (minus the linear algebra).

Coordinate-wise, one could say that a matrix is a "square" of numbers, while a tensor is a $n$-block of numbers. But this is horrible, not insightful and even a bit wrong, since those coordinates must "change in appropriate ways" (this is part of why this is horrible).

It may be best to think as follows: given a vector space $V$, a matrix can be seen in an adequate way as a bilinear map $V^* \times V \rightarrow \mathbb{R}$ (since you asked for it, I'll not enter in details. Here, $V^*$ is the dual of $V$). A tensor can be interpreted as a multilinear map $V^* \times... \times V^* \times V \times ... \times V \rightarrow \mathbb{R}$ (not necessarily the same quantity of $V^*$'s and $V$'s).

Hence, a matrix is a kind of tensor. But tensors are more general.

• instead of horrible adjective, use little more complex : P – janmarqz Sep 23 '15 at 21:50

A rank 0 tensor is a scalar.

A rank 1 tensor is a row or column vector.

A rank 2 tensor is a matrix, often square.

A rank 3 tensor? Think 3D matrix. Instead of a rectangle with data entries for each column and row, think of a cube.

Rank 4... go 4D!

• So is it correct to think of a tensor as a sort of generalized matrix in the sense that each element of a tensor could represent a scalar, vector, matrix, or "higher dimensional matrix," depending on the rank? How would you identify an element of, say, a rank 3 tensor using index notation? – Life_student Sep 21 '15 at 2:17
• I think a better way of wording my question in the comment above is this: Is a matrix just a special case of a tensor? – Life_student Sep 21 '15 at 2:18
• For practical purposes, yes, matrices are a type of tensor. The idea of a tensor generalizes the idea of matrices. – suneater Sep 21 '15 at 2:19
• A column vector can be written as $v_j$, where we select $j=1,2,3,...$ to denote a particular element. A matrix $A_{ij}$ often uses $i$ to represent the row and $j$ the column. A rank 3 tensor would look like $X_{ijk}$. If you've seen the Levi-Civita tensor, you've dealt with a rank 3 object. – suneater Sep 21 '15 at 2:22
• Take what I posted as practical advice, though. Tensors are used to facilitate coordinate transformations, and the way various objects transform can differ (you may have heard the terms covariant and contravariant in the context of general relativity). The notation changes as well, to incorporate both super and sub scripts. I'm sure there's a wealth of info around about this, but for now, I think considering tensors as generalized matrices works fine. – suneater Sep 21 '15 at 2:34

Matrices are a special type of tensor, rank 2. Scalars, vectors, matrices, are all tensors.

Honestly tensors are so general the vast majority of things you deal with in your class are tensors.