Rigid Systems of Points 77Remembering that our goal is an equation of the type (2.2.27), we have to
continue the investigation of the moment of inertia.
Let us forget for a moment that we are dealing with a rotating system,
and instead concentrate on the matrix (2.2.32). We know from linear al-
gebra that every change of the coordinate system changes the look of the
matrix; see Exercise 8.1.4, page 453 for a brief summary. For many pur-
poses, including ours, the matrix looks the best when diagonal, that is,
has zeros everywhere except on the main diagonal. While not all matrices
can have this look, every symmetric matrix is diagonal in the basis of its
normalized eigenvectors; see Exercise 8.1.5 on page 454.
By (2.2.30) and (2.2.32), the matrix ICM is symmetric, and therefore
there exists a cartesian coordinate system (?, j, k) in which ICM has at
most three non-zero entries 1^, I22 > -^33 > and all other entries zero. In other
words, there exists an orthogonal matrix U» so that the matrix ICM =
UICMUJ is diagonal. The matrix [/ is the representation in the basis
(2, j, k) of a linear transformation (rotation) that moves the vectors i, j, k
to ?, j, k, respectively. The vectors i, j, k are called the principal
axes of the system S. We will refer to the frame with the origin at the center
of mass and the basis vectors i, f, k as the principal axes frame.
Both the principal axes and the numbers 7^, J£ 2 i ^33 depend only on the
configuration of the rigid system S, that is, the positions of the point masses
rrij relative to the center of mass. A matrix can have only one diagonal look,
but in more than one basis: for example, the identity matrix looks the same
in every basis. Accordingly, the principal axes might not be unique, but
the numbers 1^, I% 2 , I^ 3 are uniquely determined by the configuration of the
system, and, in particular, do not depend on time. We know from linear
algebra that the numbers 1^, I^\,I33 are the eigenvalues of the matrix ICM >
and the matrix U* consists of the corresponding eigenvectors.
EXERCISE 2.2.10.A Given an example of a symmetric 3x3 matrix whose
entries depend on time, but whose eigenvalues do not. Can you think of a
general method for constructing such a matrix?
Formulas (2.2.30) for the entries of the matrix of inertia are true in
every basis, and therefore can be used to compute the numbers 1^, I22, 7| 3.
Since these numbers do not depend on time, the coordinates x-, yj, z* of
rrij in the principal axes frame should not depend on time either. In other
words, the principal axes frame is not rotating relative to the system S, but
is fixed in S and rotates together with S.