Moments and Principal Axes
The matrix of inertia being a real matrix (whose
elements consist entirely of real numbers) and a symmetric matrix,
there exists an orthonormal basis of vectors
in this matrix of inertia.
The principal axes are defined by vectors
and inertia principal moments are expressed by
Note:
is an orthonormal direct basis.
Expression in Any Axis System
I is the matrix of inertia with respect to orthonormal
basis Oxyz.
Huygen's theorem is used to transform the matrix
of inertia:
(parallel axis theorem).
Let I' be the matrix of inertia with respect to
orthonormal basis Pxyz where
M = {u,v,w}: transformation matrix from basis (Pxyz)
to basis (Puvw)
TM is the transposed matrix of matrix M.
J is the matrix of inertia with respect to an orthonormal
basis Puvw:
J = TM.I'.M