Question

Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified:

Are matrices and second rank tensors the same thing?

If the answer to 1 is yes, then can we think of a 3rd rank tensor as an ordered set of numbers in 3D lattice (just in the same way as we can think of a matrix as an ordered set of numbers in 2D lattice)?

Answer 1

Matrices are often first introduced to students to represent linear transformations taking vectors from Rn and mapping them to vectors in Rm. A given linear transformation may be represented by infinitely many different matrices depending on the basis vectors chosen for Rn and Rm, and a well-defined transformation law allows one to rewrite the linear operation for each choice of basis vectors.

Second rank tensors are quite similar, but there is one important difference that comes up for applications in which non-Euclidean (non-flat) distance metrics are considered, such as general relativity. 2nd rank tensors may map not just Rn to Rm, but may also map between the dual spaces of either Rn or Rm. The transformation law for tensors is similar to the one first learned for linear operators, but allows for the added flexibility of allowing the tensor to switch between acting on dual spaces or not.

Note that for Euclidean distance metrics, the dual space and the original vector space are the same, so this distinction doesn't matter in that case.

Moreover, 2nd rank tensors can act not just as maps from one vector space to another. The operation of tensor "contraction" (a generalization of the dot product for vectors) allows 2nd rank tensors to act on other second rank tensors to produce a scalar. This contraction process is generalizable for higher dimensional tensors, allowing for contractions between tensors of varying ranks to produce products of varying ranks.

To echo another answer posted here, a 2nd rank tensor at any time can indeed be represented by a matrix, which simply means rows and columns of numbers on a page. What I'm trying to do is offer a distinction between matrices as they are first introduced to represent linear operators from vector spaces, and matrices that represent the slightly more flexible objects I've described