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First, I basically just want to know what exactly what is the difference between tensors and matrices?

Second, looking for clarification on the following two (possibly confused) thoughts) of my own on this, which follow here.

(1) I was basically taught in school that a tensor is any object that transforms as:

$T_{\alpha \beta }^{'}\rightarrow \frac{\partial x _{\alpha }^{'}}{\partial x _{\alpha }} \frac{\partial x _{}\beta ^{'}}{\partial x _{}\beta } T_{\alpha \beta }$

However, is it correct that not all matrices follow this transformation law? Meaning all tensors can be represented as matrices, but not all matrices are necessarily tensors? (Hence, THIS is the difference between matrices and tensors?)

However, if that is true, then it seems a little loose to say matrices are tensors of rank 2, if indeed all matrices do not actually transform as tensors, and therefore are not tensors?

(2) Is it correct that one of things the law of general covariance says is that if a tensor equation is true in one frame then it is true in all frames? Secondly, is this made possible by the tensor transformation law?

EthanT
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  • The answer here is probably helpful to you: https://math.stackexchange.com/questions/553762/tensors-matrices-covariance-contravariance?rq=1 – Callus - Reinstate Monica Jul 03 '17 at 17:12
  • Also related: https://math.stackexchange.com/questions/412423/differences-between-a-matrix-and-a-tensor, https://math.stackexchange.com/questions/2316723/tensors-in-the-context-of-engineering-mechanics-can-they-be-explained-in-an-int, https://math.stackexchange.com/questions/10282/an-introduction-to-tensors – Hans Lundmark Jul 03 '17 at 17:16

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Short answer:

A linear map is the primary object of study in linear algebra. A matrix is a way of representing this map. A tensor is a generalization of a linear map, from which you may extract different representations corresponding to different coordinate choices, in exactly the same way as a choice of basis gives a different representation of a linear map.

Long answer can be found here.

A. Thomas Yerger
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