We can use the transformation properties of the inertia
tensor (2.24) to show that there are
special rectangular coordinate systems for which the inertia tensor
*I*' is diagonal, that is, *I*'_{ij} = 0 for *i* ne *j*. Let's assume
that ** I**' is diagonal and solve for the rotation matrix

We can examine pieces of this matrix equation by multiplying on the
right by a trivial column vector that picks out a particular column.
So we multiply on the right by the column matrix representation
*e*_{i} of each of the coordinate unit vectors _{i}.
These column matrices have a one in the *i*th row and zeros
otherwise. Using *e*_{i}' = *R**e*_{i}, we find

So, from equations (2.26) and (2.27), we have

which we recognize as an equation for the eigenvalue *I*'_{ii} and
*e*_{i}', the column matrix of components of the associated
eigenvector.

From *e*_{i}' = *R**e*_{i}, we see that the
*e*_{i}' are the columns of the rotation matrix ** R**. Now,
rotation matrices are orthogonal, so

If a matrix is real and symmetric then the eigenvalues are real.
Furthermore, if the eigenvalues are distinct then the eigenvectors are
orthogonal. However, if the eigenvalues are not distinct then the
directions of the eigenvectors for the degenerate eigenvalues are not
uniquely determined -- we have the freedom to choose particular
*e*_{i}' that are orthogonal.^{8} The linearity of
equation (2.28) implies the *e*_{i}' can be
normalized. Thus whether or not the eigenvalues are distinct we can
obtain an orthonormal set of *e*_{i}. This is enough to
reconstruct a rotation matrix ** R** that does the job we asked of
it: to rotate the coordinate system to a configuration such that the
inertia tensor is diagonal. If the eigenvalues are not distinct, the
rotation matrix

The eigenvectors and eigenvalues are determined by the requirement
that the inertia tensor be diagonal with respect to the rotated
coordinate system. Thus the rotated coordinate system has a special
orientation with respect to the body. The basis vectors _{i}'
therefore actually point along particular directions in the body. We
define the axes in the body through the center of mass with these
directions to be the *principal axes*. With respect to the coordinate
system defined by _{i}', the inertia tensor is diagonal, by construction, with the
eigenvalues *I*'_{ii} on the diagonal. Thus the moments of inertia
about the principal axes are the eigenvalues *I*'_{ii}. We call the
moments of inertia about the principal axes the *principal moments
of inertia*.

For convenience, we often label the principal moments of inertia
according to their size: *A* __<__ *B* __<__ *C*, with principal axis unit
vectors hata, , , respectively. The positive
direction along the principal axes can be chosen so that hata,
, form a right-handed rectangular coordinate basis.

Let ** x** represent the matrix of components of a vector
with respect to the basis vectors

This makes sense because the columns of ** R** are the components
of

**Exercise 2.5.** **A constraint on the moments of inertia**

Show that the sum of any two of the moments of inertia is
greater than or equal to the third moment of inertia. You may assume
the moments of inertia are with respect to orthogonal axes.

**Exercise 2.6.** **Principal moments of inertia**

For each of the configurations described below find the principal
moments of inertia with respect to the center of mass; find the
corresponding principal axes.

**a**. A regular tetrahedron consisting of four equal point masses
tied together with rigid massless wire.

**b**. A cube of uniform density.

**c**. Five equal point masses rigidly connected by massless
stuff. The point masses are at the rectangular coordinates:

**Exercise 2.7.** **This book**

Measure this book. You will admit that it is pretty dense. Don't
worry, you will get to throw it later. Show that the principal axes
are the lines connecting the centers of opposite faces of the
idealized brick approximating the book. Compute the corresponding
principal moments of inertia.

^{8} If two eigenvalues are not
distinct then linear combinations of the associated eigenvectors are
eigenvectors. This gives us the freedom to find linear combinations
of the eigenvectors that are orthonormal.