Parallel Axes Rule - Moment of Inertia Matrix - Identities For A Skew-symmetric Matrix

Identities For A Skew-symmetric Matrix

In order to compare formulations of the parallel axis theorem using skew-symmetric matrices and the tensor formulation, the following identities are useful.

Let be the skew symmetric matrix associated with the position vector R=(x, y, z), then the product in the inertia matrix becomes

$-= -\begin{bmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{bmatrix}^2 = \begin{bmatrix} y^2+z^2 & -xy & -xz \\ -y x & x^2+z^2 & -yz \\ -zx & -zy & x^2+y^2 \end{bmatrix}.$

This product can be computed using the matrix formed by the outer product using the identify

$-^2 = |\mathbf{R}|^2 -= \begin{bmatrix} x^2+y^2+z^2 & 0 & 0 \\ 0& x^2+y^2+z^2 & 0 \\0& 0& x^2+y^2+z^2 \end{bmatrix}- \begin{bmatrix}x^2 & xy & xz \\ yx & y^2 & yz \\ zx & zy & z^2\end{bmatrix},$

where is the 3x3 identify matrix.

Also notice, that

where tr denotes the sum of the diagonal elements of the outer product matrix, known as its trace.

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