= Principal moment of inertia
{title2=$I_i$}
= Principal moments of inertia
{synonym}
The <principal moments of inertia> about a specified point $O$ are the <eigenvalues> $I_1,I_2,I_3$ of the <inertia tensor> $\mathbf I_O$. Their associated <unit vectors> that are <eigenvectors> specify <principal axes> through $O$. The <spectral theorem for real symmetric matrices> guarantees an <orthonormal basis> of these <eigenvectors>, even for an irregular <rigid body>. Choose their signs to make a right-handed basis and collect them as the columns of an <orthogonal matrix> $Q$:
$$
\mathbf I_O\mathbf e_i=I_i\mathbf e_i,
\qquad Q^T\mathbf I_OQ=\operatorname{diag}(I_1,I_2,I_3).
$$
Thus <diagonalization> removes the off-diagonal coupling in the <rotational kinetic energy>. The <eigenvalues> belong to the <mass> distribution and reference point, independently of which <orthonormal basis> initially describes them. <Newtonian Dynamics by Richard Fitzpatrick/Principal Axes of Rotation>{ref}
In this <principal body frame>, write <angular velocity> as $\boldsymbol\omega=\sum_i\omega_i\mathbf e_i$. Then
$$
\mathbf L_O=\sum_i I_i\omega_i\mathbf e_i,
\qquad
K=\frac12\sum_i I_i\omega_i^2.
$$
Use these as total <angular momentum> and <kinetic energy> only under the reference-point conditions stated under <inertia tensor>. Along a single <principal axis> with $I_i>0$, <angular momentum> is parallel to <angular velocity>. For a mixture of directions with different <principal moments of inertia>, it generally is not. A <principal axis> need not be a geometric symmetry axis: it is selected by the complete <mass> distribution. A <principal axis> also need not give stable free rotation; for three distinct <principal moments of inertia>, the <intermediate axis theorem> supplies the stability distinction.
For any <unit vector> $\mathbf n=\sum_i n_i\mathbf e_i$,
$$
I_{O,\mathbf n}=\sum_i I_i n_i^2,
\qquad \sum_i n_i^2=1,
\qquad \min_i I_i\leq I_{O,\mathbf n}\leq\max_i I_i.
$$
The axial <moment of inertia> is a weighted average of the <principal moments of inertia>. At fixed <angular speed>, the least and greatest <rotational kinetic energies> occur in the corresponding <eigenspaces>. At fixed nonzero <angular momentum> magnitude $L$, instead, $K=\tfrac12\sum_i L_i^2/I_i$, so the largest <principal moment of inertia> gives the smallest <rotational kinetic energy>. These comparisons hold the reference point fixed.
There is an additional physical restriction beyond <positive semidefiniteness>. In <principal axes> coordinates,
$$
I_1+I_2-I_3=2\int z^2\,dm\geq0,
$$
with two analogous cyclic inequalities. Therefore three arbitrary positive numbers do not always represent a possible <inertia tensor>. For a distribution in the $xy$ plane through $O$, $I_3=I_1+I_2$, as in the planar <inertia tensor of paired point masses>. A uniform solid <ellipsoid> of <mass> $M$ and semiaxes $a,b,c$ supplies a continuous three-dimensional example: using its <mass second-moment tensor>, its central <principal moments of inertia> are $M(b^2+c^2)/5$, $M(a^2+c^2)/5$, $M(a^2+b^2)/5$. Setting $a=b=c=R$ recovers the <moment of inertia of a uniform solid sphere>.
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