Off-Diagonal Terms in Moment of Inertia Tensor

This all makes sense, but what about those $-1$ off-diagonal terms in $\mathbf{I}$ ? Consider the vector angular momentum (§B.4.14):

$$\underline{L}\eqsp \mathbf{I}\,\underline{\omega}\eqsp m\left[\begin{array}{ccc} I_{11} & I_{12} & I_{13}\\ [2pt] I_{21} & I_{22} & I_{23}\\ [2pt] I_{31} & I_{32} & I_{33} \end{array}\right] \left[\begin{array}{c} \omega_1 \\ [2pt] \omega_2 \\ [2pt] \omega_3\end{array}\right]$$

We see that the off-diagonal terms $I_{ij}$ correspond to a coupling of rotation about $\underline{e}_i$ with rotation about $\underline{e}_j$ . That is, there is a component of moment-of-inertia $I_{ij}$ that is contributed (or subtracted, as we saw above for $\underline{\omega}=[1,1,0]^T$ ) when both $\omega_i$ and $\omega_j$ are nonzero. These cross-terms can be eliminated by diagonalizing the matrix [452],^B.25as discussed further in the next section.