The representation of the rotational kinetic energy in terms of the
inertia tensor was derived with the help of a rectangular coordinate
system with basis vectors _{i}. There was nothing special
about this particular rectangular basis. So, the kinetic energy must
have the same form in any rectangular coordinate system. We can use
this fact to derive how the inertia tensor changes if the body or the
coordinate system is rotated.

Let's talk a bit about *active* and *passive* rotations.
The rotation of the vector by the rotation *R* produces a
new vector ' = *R* . We may write in terms of
its components with respect to some arbitrary rectangular coordinate
system with orthonormal basis vectors _{i}: = *x*^{0}
_{0} + *x*^{1} _{1} + *x*^{2} _{2}. Let ** x** indicate
the column matrix of components

Alternately, we can rotate the coordinate system by rotating the basis
vectors, but leave other vectors that might be represented in terms of
them unchanged. If a vector is unchanged but the basis vectors are
rotated, then the components of the vector on the rotated basis vectors
are not the same as the components on the original basis vectors.
Denote the rotated basis vectors by _{i}' = *R* _{i}. The
component of a vector along a basis vector is the dot product of the
vector with the basis vector. So the components of the vector
along the rotated basis _{i}' are (*x*')^{i} =
· _{i}' = · (*R* _{i}) = (*R*^{-1} )
· _{i}.^{5}
Thus the components with
respect to the rotated basis elements are the same as the components
of the rotated vector *R*^{-1} with respect to the original
basis. In terms of components, if the vector has components
** x** with respect to the original basis vectors

With respect to the rectangular basis _{i} the rotational
kinetic energy is written

In terms of matrix representations, the kinetic energy is

where is the column of components representing
.^{6}
If we rotate the coordinate system by the passive
rotation *R* about the center of rotation, the new basis vectors are
_{i}' = *R* _{i}. The components ' of the vector
with respect to the rotated coordinate system satisfy

where ** R** is the matrix representation of

However, if we had started with the basis _{i}', we would have
written the kinetic energy directly as

where the components are taken with respect to the _{i}' basis.
Comparing the two expressions, we see that

Thus the inertia matrix transforms by a similarity
transformation.^{7}

^{4} An orthogonal matrix ** R** satisfies

^{5} The last equality follows from the fact
that the rotation of two vectors preserves the dot product:
· = (*R*) · (*R*), or (*R*^{-1} )
· = · (*R* ).

^{6} We take a 1-by-1 matrix as a number.

^{7} That the inertia tensor transforms in this
manner could have been deduced from its
definition (2.14). However, it seems that the
argument based on the coordinate-system independence of the
kinetic energy provides insight.