The inertia tensor is called a rank two tensor because it has two indices. It illustrates the difference between a tensor and a matrix. Because the inertia tensor depends on the coordinates in a clear way, we can write down how it must behave under rotations. If we rotate the coordinate system, the and must be transformed with a rotation matrix. The is invariant since it is a dot product. So lets try the transformation
In summary, the inertia tensor transforms under rotations like any other rank 2 tensor.
Jim Branson 2012-10-21