### Principal Axes of Rotation

A *principal axis of rotation* (or *principal direction*) is
an eigenvector of the *mass moment of inertia tensor* (introduced
in the previous section) defined relative to some point (typically the
center of mass). The corresponding eigenvalues are called the
*principal moments of inertia*.
Because the moment of inertia tensor is defined relative to the point
in the space, the principal axes all pass through that point
(usually the center of mass).

As derived above (§B.4.14), the angular momentum vector is given by the moment of inertia tensor times the angular-velocity vector:

*principal moment of inertia*. If we set the rigid body assocated with rotating about the axis , then is the mass moment of inertia of the body for that rotation. As will become clear below, there are always three mutually orthogonal principal axes of rotation, and three corresponding principal moments of inertia (in 3D space, of course).

#### Positive Definiteness of the Moment of Inertia Tensor

From the form of the moment of inertia tensor introduced in Eq.(B.24)

*symmetric*. Moreover, for any normalized angular-velocity vector we have

since
is unit length, and projecting it onto any other vector
can only shorten it or leave it unchanged. That is,
, with equality occurring for
for any nonzero
. Zooming out,
*of course* we expect any moment of inertia for a positive
mass to be nonnegative. Thus,
is *symmetric
nonnegative definite*. If furthermore
and
are not
collinear, *i.e.*, if there is any nonzero angle between them, then
is *positive definite* (and ). As is well known in
linear algebra [329], real, symmetric, positive-definite
matrices have *orthogonal eigenvectors* and *real, positive
eigenvalues*. In this context, the orthogonal eigenvectors are
called the *principal axes of rotation*. Each corresponding
eigenvalue is the moment of inertia about that principal axis--the
corresponding principal moment of inertia. When angular velocity
vectors
are expressed as a linear combination of the principal
axes, there are no cross-terms in the moment of inertia tensor--no
so-called *products of inertia*.

The three principal axes are *unique* when the eigenvalues of
(principal moments of inertia) are *distinct*. They are
not unique when there are repeated eigenvalues, as in the example
above of a disk rotating about any of its diameters
(§B.4.4). In that example, one principal
axis, the one corresponding to eigenvalue , was
(*i.e.*,
orthogonal to the disk and passing through its center), while any two
orthogonal diameters in the plane of the disk may be chosen as the
other two principal axes (corresponding to the repeated eigenvalue
).

Symmetry of the rigid body about any axis
(passing through the
origin) means that
is a principal direction. Such a symmetric
body may be constructed, for example, as a *solid of
revolution*.^{B.26}In rotational dynamics, this case is known as the *symmetric top*
[270]. Note that the center of mass will lie
somewhere along an axis of symmetry. The other two principal axes can
be arbitrarily chosen as a mutually orthogonal pair in the (circular)
plane orthogonal to the
axis, intersecting at the
axis. Because of the circular symmetry about
, the two
principal moments of inertia in that plane are equal. Thus the moment
of inertia tensor can be diagonalized to look like

**Next Section:**

Rotational Kinetic Energy Revisited

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Mass Moment of Inertia Tensor