I have calculated the moment of inertia tensor of a cube about its center of mass: $I=\dfrac{1}{6}Mb^2\{1\}$ where $\{1\}$ is the identity matrix. So the principal moments of inertia are all 1 (1 is an eigenvalue of multiplicity 3).
When you attempt to calculate the eigenvectors to identify the principal axes, the result is that every vector is an eigenvector of the identity matrix, which is expected.
My textbook, however, states, "Thus, we find that, for the choice of the origin at the center of mass of the cube, the principal axes are perpendicular to the faces of the cube."
Why is this true? I thought that the principal axes were simply the eigenvectors of the diagonalized matrix, which would mean that every axis that passes through the center of mass of a cube is a principal axis.
Best Answer
It is simply not true, as stated.
A cube is as perfectly balanced around its center of mass as a sphere is. You have shown it mathematically, and you are perfectly correct.
However, the principal axes may be chosen perpendicular to the faces of the cube. Of course, when the point is the center of mass, this is no better than any other choice, except for considerations of symmetry and mathematical convenience.