- #1

Filip Larsen

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- Summary:
- Do rotational invariance of the vector cross product also carry over to the cross product matrix operator?

Given that the normal vector cross product is rotational invariant, that is $$\mathbf R(a\times b) = (\mathbf R a)\times(\mathbf R b),$$ where ##a, b \in \mathbb{R}^3## are two arbitrary (column) vectors and ##\mathbf R## is a 3x3 rotation matrix, and given the cross product matrix operator defined by $$ \left[a\right]_\times = \begin{bmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{bmatrix} ,$$ such that ##a \times b = \left[a\right]_\times b##, my question now is if rotational invariance also applies for this operator, that is if it in general holds that $$\mathbf R \left[ a \right]_\times \stackrel{?}{=} \left[ \mathbf R a \right]_\times \mathbf R$$ Specifically for my current use, with ##\mathbf C## being 3x3 (positive semi-definite) matrix and utilizing ##\mathbf R^{-1} = \mathbf R^T## holds for a rotation matrix can I then conclude that ## \left[\mathbf R a \right]_\times \mathbf C \left[ \mathbf R a \right]_\times^T = \left( \mathbf R \left[ a \right]_\times \mathbf R^T \right) \mathbf C \left(\mathbf R \left[ a \right]_\times \mathbf R^T \right)^T = \left( \mathbf R \left[ a \right]_\times \right) \left( \mathbf R^T \mathbf C \mathbf R \right) \left( \mathbf R \left[ a \right]_\times \right)^T## always holds, as I am inclined to believe?

My (engineering) intuition tries to tell me that since the relation with the question marks holds when applied to a column vector, due to ##\left( \mathbf R \left[ a \right]_\times \right) b = \mathbf R \left( a\times b \right) = \left(\mathbf R a\right) \times \left(\mathbf R b\right) = \left( \left[ \mathbf R a \right]_\times\right) \left( \mathbf R b \right) = \left( \left[ \mathbf R a \right]_\times \mathbf R \right) b##, and since an equation involving multiplication of a 3x3 matrix can be separated into 3 equations with multiplication of each column vector of the matrix, then the relation must also hold when combined back into a general 3x3 matrix, but I worry if such math hand waving has math holes in it my engineering intuition can't see.

My (engineering) intuition tries to tell me that since the relation with the question marks holds when applied to a column vector, due to ##\left( \mathbf R \left[ a \right]_\times \right) b = \mathbf R \left( a\times b \right) = \left(\mathbf R a\right) \times \left(\mathbf R b\right) = \left( \left[ \mathbf R a \right]_\times\right) \left( \mathbf R b \right) = \left( \left[ \mathbf R a \right]_\times \mathbf R \right) b##, and since an equation involving multiplication of a 3x3 matrix can be separated into 3 equations with multiplication of each column vector of the matrix, then the relation must also hold when combined back into a general 3x3 matrix, but I worry if such math hand waving has math holes in it my engineering intuition can't see.