# Is the moment of inertia matrix a tensor?

## Homework Statement

Is the moment of inertia matrix a tensor? Hint: the dyadic product of two vectors transforms according to the rule for second order tensors.
• $I$ is the inertia matrix
• $L$ is the angular momentum
• $\omega$ is the angular velocity

## Homework Equations

The transformation rule for a second order tensor is: $I'_{ij} = C_{ip}C_{jq}I_{pq}$. A dyadic product of two vectors $u$ and $v$ is a matrix of the form: $A_{ij} = u_iv_j$.

## The Attempt at a Solution

In a previous homework, we proved that a dyadic product transforms according to the rule for second order tensors. I would like to show that the moment of inertia matrix is a dyadic product. However, we never defined it that way in class yet, so I am unsure of how exactly this would work. We have: $L_i = I_{ij}w_j$. Now, is it ok to then write that $L_i\frac{1}{w_j} = I_{ij}$, so that $I_{ij}$ is a dyadic product of $L$ and the vector whose components are $\frac{1}{w_i}$? That feels wrong to me, because dividing out by something with a dummy variable like that doesn't seem like it should be valid, but I'm not actually sure!