# Moment of Inertia - Moment of Inertia Reference Frames - Identities For A Skew-symmetric Matrix

Identities For A Skew-symmetric Matrix

In order to compare formulations of the inertia matrix in terms of a product of skew-symmetric matrices and in terms of a 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

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

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.