# Ek I 2 Xp Xft i 2 [px

Thus, both the angular momentum and the kinetic energy can be expressed in terms of / and <o.

Some authors (e.g., Whittaker ) define the negatives of the off-diagonal elements

Pjk = ~ ljk as products of inertia; but other authors (e.g., Goldstein  and Kibble ) define the elements IJk, without the minus sign, as the products of inertia. Still other authors (e.g., Thomson  and Kaplan ) define the products of inertia as Whittaker does, but denote them by ljk, so that the off-diagonal elements of the moment of inertia tensor are — ljk. The quantity I is called a tensor because it has specific transformation properties under a real orthogonal transformation (see, for example, Goldstein  or Synge and Schild .) It is sufficient for our purposes to think of the moment of inertia tensor as a real, symmetric 3x3 matrix. Because the moment of inertia tensor is a real, symmetric matrix, it has three real orthogonal eigenvectors and three real eigenvalues (see Appendix C) satisfying the equation

The scalars /,, /2, and /3 are the principal moments of inertia, and the unit vectors P„ P2, and P3 are the principal axes. These quantities were introduced in a more intuitive manner in Section 15.1. If we use the principal axes as the coordinate axes of a spacecraft reference frame, the moment of inertia tensor takes the diagonal form

In this coordinate frame (and only in this frame), Eqs. (16-34) and (16-35) can be expressed as