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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 momentumB.4.14):

$\displaystyle \underline{L}\eqsp \mathbf{I}\,\underline{\omega}\eqsp
...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 [449],B.25as discussed further in the next section.

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Positive Definiteness of the Moment of Inertia Tensor
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Example with Coupled Rotations