- #1
Trying2Learn
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- TL;DR Summary
- Relationship of Skew Symmetry, vectors, cross product and Lie Bracket
As an aside, fresh_42 commented and I made an error in my post that is now fixed. His comment, below, is not valid (my fault), in that THIS post is now fixed.Assume s and w are components of vectors, both in the same frame
Assume S and W are skew symmetric matrices formed from the vector components
I want to prove this (that I CAN commute the first two terms on each side)
W*S*W*s = S*W*W*s
Thus, I want to prove
W*S*W*s - S*W*W*s = 0
(W*S-S*W)*W*s =0
However, the parenthetical is the Lie Bracket. Thus
[W,S]*W*s = 0
So, is the above, true?
I can show it works for any general skew symmetric system, using the symbolic manipulator in Matlab
But can someone explain why this is true?
Assume S and W are skew symmetric matrices formed from the vector components
I want to prove this (that I CAN commute the first two terms on each side)
W*S*W*s = S*W*W*s
Thus, I want to prove
W*S*W*s - S*W*W*s = 0
(W*S-S*W)*W*s =0
However, the parenthetical is the Lie Bracket. Thus
[W,S]*W*s = 0
So, is the above, true?
I can show it works for any general skew symmetric system, using the symbolic manipulator in Matlab
But can someone explain why this is true?
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