# Moment of inertia as a tensor

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.

• Do you know what a 3×3 matrix is? – ja72 Sep 11 '14 at 17:28
• @ja72 Ofcourse . – Žan Žurič Sep 11 '14 at 17:54
• If you like this question you may also enjoy reading this and this Phys.SE posts. – Qmechanic Sep 11 '14 at 17:58

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