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).

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Qualitatively (or mathematically &quot;light&quot;), 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).

vard6vv · Accepted Answer

Coordinate-wise, one could say that a matrix is a &quot;square&quot; 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 &quot;change in appropriate ways&quot; (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    ∗    ×  V  →      R   (since you asked for it, I&#039;ll not enter in details. Here,       V    ∗   is the dual of   V). A tensor can be interpreted as a multilinear map       V    ∗    ×  .  .  .  ×      V    ∗    ×  V  ×  .  .  .  ×  V  →      R   (not necessarily the same quantity of       V    ∗  &#039;s and   V&#039;s).Hence, a matrix is a kind of tensor. But tensors are more general.

Davian Maynard · Accepted Answer

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.

Qualitatively (or mathematically "light"), could someone describe the difference between a matrix an

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