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.


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... 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 This product can be computed using the matrix formed ... the sum of the diagonal elements of the outer product matrix, known as its trace ...

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