Moment of inertia as a tensor

Question

A professor at my university briefly stated that moment of inertia is a tensor and can be represented by a $3×3$ matrix. I don't have a good idea of what a tensor is, so I would be grateful if someone could explain how to intuitively think of moment of inertia as a tensor.

Answer 1

The moment of inertia tensor contains elements which relate one component of angular velocity to another component of angular momentum.

$$ \boldsymbol{L} = \mathbf{I}\,\boldsymbol{\omega} $$ $$ \begin{pmatrix} L_x \\ L_y \\ L_z \end{pmatrix} = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{xy} & I_{yy} & -I_{yz} \\ I_{xz} & I_{yz} & I_{zz} \end{bmatrix} \begin{pmatrix} \omega_x \\ \omega_y \\ \omega_z \end{pmatrix}$$

So, for example, the element $I_{xz}$ relates the speed $\omega_x$ with the momentum $L_z$ and since it is always a symmetric tensor, the speed $\omega_z$ with the momentum $L_x$. If the case was that $I_{xz}=0$ then $L_z$ does not contain a component due to $\omega_x$ and vice versa.

Also see here for a post on how to rotate a mass moment of inertia tensor from the local (body) coordinate system to the inertial (world) coordinate system. Then look at this for the full dynamics of rigid bodies.