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