Editing Moment of inertia (section)

=== Principal axes ===
Measured in the body frame, the inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the [[eigendecomposition of a matrix|eigendecomposition]] into the product of a rotation matrix <math>\mathbf{Q}</math> and a diagonal matrix <math>\boldsymbol{\Lambda}</math>, given by
<math display="block">\mathbf{I}_\mathbf{C}^B = \mathbf{Q}\boldsymbol{\Lambda}\mathbf{Q}^\mathsf{T},</math>
where
<math display="block">\boldsymbol{\Lambda} = \begin{bmatrix}
  I_1 & 0 & 0 \\
  0 & I_2 & 0 \\
  0 & 0 & I_3
\end{bmatrix}.</math>

The columns of the rotation matrix <math>\mathbf{Q}</math> define the directions of the principal axes of the body, and the constants <math>I_1</math>, <math>I_2</math>, and <math>I_3</math> are called the '''principal moments of inertia'''. This result was first shown by [[James Joseph Sylvester|J. J. Sylvester (1852)]], and is a form of [[Sylvester's law of inertia]].<ref name=syl852>{{cite journal |author=Sylvester, J J | title=A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares | journal=Philosophical Magazine |series=4th Series| volume=4 | issue=23 | pages=138–142 | year=1852 |  url=http://www.maths.ed.ac.uk/~aar/sylv/inertia.pdf | doi= 10.1080/14786445208647087 | access-date=June 27, 2008}}</ref><ref name=norm>{{cite book| author=Norman, C.W.| title=Undergraduate algebra | publisher=[[Oxford University Press]] | pages=360–361 | year=1986 |  isbn=0-19-853248-2 }}</ref> When the body has an axis of symmetry (sometimes called the '''figure axis''' or '''axis of figure''') then the other two moments of inertia will be identical and any axis perpendicular to the axis of symmetry will be a principal axis.

A toy [[Spinning top|top]] is an example of a rotating rigid body, and the word ''top'' is used in the names of types of rigid bodies.  When all principal moments of inertia are distinct, the principal axes through [[center of mass]] are uniquely specified and the rigid body is called an '''asymmetric top'''. If two principal moments are the same, the rigid body is called a '''symmetric top''' and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a '''spherical top''' (although it need not be spherical) and any axis can be considered a principal axis, meaning that the moment of inertia is the same about any axis.

The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order <math>m</math>, meaning it is symmetrical under rotations of {{math|[[turn (geometry)|360°]]/''m''}} about the given axis, that axis is a principal axis. When <math>m > 2</math>, the rigid body is a symmetric top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, for example, a cube or any other [[Platonic solid]].

The [[motion (physics)|motion]] of [[vehicle]]s is often described in terms of [[yaw, pitch, and roll]] which usually correspond approximately to rotations about the three principal axes. If the vehicle has bilateral symmetry then one of the principal axes will correspond exactly to the transverse (pitch) axis.

A practical example of this mathematical phenomenon is the routine automotive task of [[Tire balance|balancing a tire]], which basically means adjusting the distribution of mass of a car wheel such that its principal axis of inertia is aligned with the axle so the wheel does not wobble.

[[Rotational spectroscopy#Classification of molecular rotors|Rotating molecules are also classified]] as asymmetric, symmetric, or spherical tops, and the structure of their [[Rotational spectroscopy|rotational spectra]] is different for each type.