What are the principal moments of inertia? (In relation to eigenvalues, eigenvectors and point masses) I don't have an actual question however I would like know and understand how to calculate the principal moment of inertia of a mass. What is the principal moment of inertia? 
To find it, do I need eigenvalues and eigenvectors?
For example, if there were two unit masses located at (x, y, z) and (x, y, z). Where the eigenvalues determined are 1 and 2. And the eigenvectors are (-1, -1, 1)' and (0, 0, 1)'. (These are entirely made up to aid in my explanation).
Any thoughts and knowledge on this topic would be greatly appreciated. I apologise for how vague my question is.
 A: First, calculate the moment of inertia tensor $\mathbf{I}$. This is done by integrating the moment of inertia of each mass element about the origin, as follows
$$
\mathbf{I} = \int \left(r^2\mathbf{E} - \mathbf{r}\mathbf{r}\right)\rho(\mathbf{r})d^3\mathbf{r}
$$
where $\mathbf{E}$ is the identity tensor and $\rho(\mathbf{r})$ is the mass density. If you simply have a collection of point masses $m_i$ at locations $\mathbf{r}_i$, this integral reduces to a sum over those masses:
$$
\mathbf{I} = \sum_i \left(r_i^2\mathbf{E}-\mathbf{r}_i\mathbf{r}_i\right)m_i.
$$
So how does $\mathbf{I}$ relate to the principal axes? Well, the principal axes are the axes of rotation where the angular momentum $\mathbf L$ is parallel to the angular velocity $\boldsymbol\omega$. By construction, the moment of inertia tensor satisfies
$$
\mathbf{L} = \mathbf{I}\cdot\boldsymbol\omega
$$
which immediately tells us that the principal axes are simply the eigenvectors of $\mathbf{I}$, and the moment of inertia about each of those axes is the corresponding eigenvalue.
If you need to know how to find the eigenvectors of an arbitrary matrix, there should be plenty of answers around here that relate to that.
