Re: Let's Play 1 2 3...
**The moment of inertia tensor about the center of mass of a 3 dimensional rigid body is related to the covariance matrix of a trivariate random vector whose probability density function is proportional to the pointwise density of the rigid body by
I=n (\mathbf{1}_{3 imes 3} \operatorname{tr}(\Sigma) - \Sigma)
where n is the number of points.
The structure of the moment-of-inertia tensor comes from the fact that it is to be used as a bilinear form on rotation vectors in the form
\frac{1}{2}\omega^T I \omega.\,\!
Each element of mass has a kinetic energy of
\frac{1}{2} m |v|^2. \,\!
The velocity of each element of mass is \omega imes r where r is a vector from the center of rotation to that element of mass. The cross product can be converted to matrix multiplication so that
\omega imes r = [r]_ imes^T \omega\,\!
and similarly
(\omega imes r)^T = ([r]_ imes \omega)^T = \omega^T [r]_ imes^T. \,\!
Thus
|v|^2 = (\omega imes r)^T(\omega imes r)=\omega^T [r]_ imes^T [r]_ imes \omega \,\!
plugging in the definition of \cdot]_ imes the [r]_ imes^T [r]_ imes term leads directly to the structure of the moment tensor.
**