Distributive property of a matrix on a cross product
$begingroup$
Let $vec x, vec y in mathbb{R}^3$ and $bf A $ be a $3 times 3$ real matrix. Under what conditions does $bf A$ distribute over a cross product:
$$ mathbf{A} (vec x times vec y) = (mathbf{A}vec x) times (mathbf{A} vec y) $$
The cross product in this case is the vanilla one over $mathbb{R}^3$:
$$ (vec a times vec b)_i = epsilon_{ijk} a_j b_k$$
Where $epsilon$ is the Levi-Civita symbol. My suspicion is matrices where $mathbf{A}^{-1} = mathbf{A}^T$ like the rotations in $mathbb{R}^3$ would satisfy this condition. I can imagine the equality holding when $vec x$ is aligned with the axis of rotation (i.e. $mathbf{A}vec x = vec x$) and therefore rotating $vec y$ also rotates $vec x times vec y$.
Would anyone back the general case up with a proof for me?
matrices rotations cross-product
asked Dec 13 '18 at 22:48
cmscms
101
$endgroup$
add a comment |
$begingroup$
Let $vec x, vec y in mathbb{R}^3$ and $bf A $ be a $3 times 3$ real matrix. Under what conditions does $bf A$ distribute over a cross product:
$$ mathbf{A} (vec x times vec y) = (mathbf{A}vec x) times (mathbf{A} vec y) $$
The cross product in this case is the vanilla one over $mathbb{R}^3$:
$$ (vec a times vec b)_i = epsilon_{ijk} a_j b_k$$
Where $epsilon$ is the Levi-Civita symbol. My suspicion is matrices where $mathbf{A}^{-1} = mathbf{A}^T$ like the rotations in $mathbb{R}^3$ would satisfy this condition. I can imagine the equality holding when $vec x$ is aligned with the axis of rotation (i.e. $mathbf{A}vec x = vec x$) and therefore rotating $vec y$ also rotates $vec x times vec y$.
Would anyone back the general case up with a proof for me?
matrices rotations cross-product
asked Dec 13 '18 at 22:48
cmscms
101
$endgroup$
$begingroup$
Of course immediately after posting I found this question here: math.stackexchange.com/questions/279173/… Should I delete this question or allow it to me marked duplicate?
$endgroup$
– cms
Dec 13 '18 at 22:58
$begingroup$
Yes, absolutely, DELETE!
$endgroup$
– Cosmas Zachos
Jan 4 at 2:22
add a comment |
$begingroup$
Let $vec x, vec y in mathbb{R}^3$ and $bf A $ be a $3 times 3$ real matrix. Under what conditions does $bf A$ distribute over a cross product:
$$ mathbf{A} (vec x times vec y) = (mathbf{A}vec x) times (mathbf{A} vec y) $$
The cross product in this case is the vanilla one over $mathbb{R}^3$:
$$ (vec a times vec b)_i = epsilon_{ijk} a_j b_k$$
Where $epsilon$ is the Levi-Civita symbol. My suspicion is matrices where $mathbf{A}^{-1} = mathbf{A}^T$ like the rotations in $mathbb{R}^3$ would satisfy this condition. I can imagine the equality holding when $vec x$ is aligned with the axis of rotation (i.e. $mathbf{A}vec x = vec x$) and therefore rotating $vec y$ also rotates $vec x times vec y$.
Would anyone back the general case up with a proof for me?
matrices rotations cross-product
asked Dec 13 '18 at 22:48
cmscms
101
$endgroup$
Let $vec x, vec y in mathbb{R}^3$ and $bf A $ be a $3 times 3$ real matrix. Under what conditions does $bf A$ distribute over a cross product:
$$ mathbf{A} (vec x times vec y) = (mathbf{A}vec x) times (mathbf{A} vec y) $$
The cross product in this case is the vanilla one over $mathbb{R}^3$:
$$ (vec a times vec b)_i = epsilon_{ijk} a_j b_k$$
Where $epsilon$ is the Levi-Civita symbol. My suspicion is matrices where $mathbf{A}^{-1} = mathbf{A}^T$ like the rotations in $mathbb{R}^3$ would satisfy this condition. I can imagine the equality holding when $vec x$ is aligned with the axis of rotation (i.e. $mathbf{A}vec x = vec x$) and therefore rotating $vec y$ also rotates $vec x times vec y$.
Would anyone back the general case up with a proof for me?
matrices rotations cross-product
matrices rotations cross-product
asked Dec 13 '18 at 22:48
cmscms
101
asked Dec 13 '18 at 22:48
cmscms
101
asked Dec 13 '18 at 22:48
cmscms
101
asked Dec 13 '18 at 22:48
cmscms
101
asked Dec 13 '18 at 22:48
cmscms
101
101
$begingroup$
Of course immediately after posting I found this question here: math.stackexchange.com/questions/279173/… Should I delete this question or allow it to me marked duplicate?
$endgroup$
– cms
Dec 13 '18 at 22:58
$begingroup$
Yes, absolutely, DELETE!
$endgroup$
– Cosmas Zachos
Jan 4 at 2:22
add a comment |
$begingroup$
Of course immediately after posting I found this question here: math.stackexchange.com/questions/279173/… Should I delete this question or allow it to me marked duplicate?
$endgroup$
– cms
Dec 13 '18 at 22:58
$begingroup$
Yes, absolutely, DELETE!
$endgroup$
– Cosmas Zachos
Jan 4 at 2:22
$begingroup$
Of course immediately after posting I found this question here: math.stackexchange.com/questions/279173/… Should I delete this question or allow it to me marked duplicate?
$endgroup$
– cms
Dec 13 '18 at 22:58
$begingroup$
Of course immediately after posting I found this question here: math.stackexchange.com/questions/279173/… Should I delete this question or allow it to me marked duplicate?
$endgroup$
– cms
Dec 13 '18 at 22:58
$begingroup$
Yes, absolutely, DELETE!
$endgroup$
– Cosmas Zachos
Jan 4 at 2:22
$begingroup$
Yes, absolutely, DELETE!
$endgroup$
– Cosmas Zachos
Jan 4 at 2:22
add a comment |
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$begingroup$
Of course immediately after posting I found this question here: math.stackexchange.com/questions/279173/… Should I delete this question or allow it to me marked duplicate?
$endgroup$
– cms
Dec 13 '18 at 22:58
$begingroup$
Yes, absolutely, DELETE!
$endgroup$
– Cosmas Zachos
Jan 4 at 2:22