# Moment of Inertia Tensor Terminology

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.

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.

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