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I've learned about the moment of inertia tensor as a matrix that can be used to compute angular momentum, moment of inertia, etc. for a system. But why is it often described as a tensor instead of a matrix? I've heard that a tensor is something with each component having multiple basis vectors; for example, the rank-2 stress tensor which describes direction of stress as well as direction of face. However I'm not seeing how that fits into the moment of inertia case.

Thanks for the help.

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Under a rotation, coordinates transform like

$$x’_i=R_{ij}x_j$$

where $R$ is a rotation matrix. (A repeated index like $j$ here is implicitly summed over in this index notation.)

A vector (also known as a tensor of rank 1) consists of 3 components that transform in the same way,

$$v’_i=R_{ij}v_j.$$

A tensor of rank 2 consists of 9 components that transform like the product $v_iv_j$ of two vectors:

$$I’_{ij}=R_{ik}R_{jl}I_{kl}.$$

The moment-of-inertia tensor has this transformation law, which explains why it is called a tensor of rank 2 rather than simply a matrix. A matrix is just a square array of numbers with no particular transformation law under coordinate transformations.

A tensor of rank 3 consists of 27 components that transform like the product $v_iv_jv_k$ of three vectors:

$$T’_{ijk}=R_{il}R_{jm}R_{kn}T_{lmn},$$

and so on for any rank. So there are higher-rank tensors which don’t look like matrices.

There are fancier (and better) ways to think about tensors as more than just a bunch of components, but thinking about how their components transform under a coordinate transformation is a common first intro to them.

The tensor concept then generalizes to higher-dimensional spaces such as 4D spacetime; to other transformations besides rotations, such as Lorentz transformations; and to curved manifolds rather than flat Euclidean space and Minkowski spacetime. If you continue studying physics you are likely to encounter all of these kinds of tensors.

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  • $\begingroup$ Thank you for the explanation! $\endgroup$ – DanDan0101 Aug 16 '19 at 5:13
  • $\begingroup$ Never a response from a good user should have 0 score after a positive check. :-) $\endgroup$ – Sebastiano Aug 16 '19 at 22:35

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