Generalized rotation matrix in N dimensional space around N-2 unit vector












25












$begingroup$


There is a 2d rotation matrix around point $(0, 0)$ with angle $theta$.



$$
left[ begin{array}{ccc}
cos(theta) & -sin(theta) \
sin(theta) & cos(theta) end{array} right]
$$



Next, there is a 3d rotation matrix around point $(0, 0, 0)$ and unit axis $(u_x, u_y, u_z)$ with angle $theta$ (Rodrigues' Rotation Formula).



begin{bmatrix} cos theta +u_x^2 left(1-cos thetaright) & u_x u_y left(1-cos thetaright) - u_z sin theta & u_x u_z left(1-cos thetaright) + u_y sin theta \ u_y u_x left(1-cos thetaright) + u_z sin theta & cos theta + u_y^2left(1-cos thetaright) & u_y u_z left(1-cos thetaright) - u_x sin theta \ u_z u_x left(1-cos thetaright) - u_y sin theta & u_z u_y left(1-cos thetaright) + u_x sin theta & cos theta + u_z^2left(1-cos thetaright)
end{bmatrix}



How it is possible to generalize rotation matrix on $N$ dimension around zero point and $N-2$ dimensional unit axis with angle $theta$?






linear-algebra matrices rotations







share|cite|improve this question











$endgroup$












  • $begingroup$
    @RobertIsrael could you please tell me the reference (name of the formula, article, book, etc.) for your formula above (a matrix that rotates the span of u and v by angle theta)? Thank you in advance.
    $endgroup$
    – user71237
    Apr 6 '13 at 8:13
















25












$begingroup$


There is a 2d rotation matrix around point $(0, 0)$ with angle $theta$.



$$
left[ begin{array}{ccc}
cos(theta) & -sin(theta) \
sin(theta) & cos(theta) end{array} right]
$$



Next, there is a 3d rotation matrix around point $(0, 0, 0)$ and unit axis $(u_x, u_y, u_z)$ with angle $theta$ (Rodrigues' Rotation Formula).



begin{bmatrix} cos theta +u_x^2 left(1-cos thetaright) & u_x u_y left(1-cos thetaright) - u_z sin theta & u_x u_z left(1-cos thetaright) + u_y sin theta \ u_y u_x left(1-cos thetaright) + u_z sin theta & cos theta + u_y^2left(1-cos thetaright) & u_y u_z left(1-cos thetaright) - u_x sin theta \ u_z u_x left(1-cos thetaright) - u_y sin theta & u_z u_y left(1-cos thetaright) + u_x sin theta & cos theta + u_z^2left(1-cos thetaright)
end{bmatrix}



How it is possible to generalize rotation matrix on $N$ dimension around zero point and $N-2$ dimensional unit axis with angle $theta$?






linear-algebra matrices rotations







share|cite|improve this question











$endgroup$












  • $begingroup$
    @RobertIsrael could you please tell me the reference (name of the formula, article, book, etc.) for your formula above (a matrix that rotates the span of u and v by angle theta)? Thank you in advance.
    $endgroup$
    – user71237
    Apr 6 '13 at 8:13














25












25








25


16



$begingroup$


There is a 2d rotation matrix around point $(0, 0)$ with angle $theta$.



$$
left[ begin{array}{ccc}
cos(theta) & -sin(theta) \
sin(theta) & cos(theta) end{array} right]
$$



Next, there is a 3d rotation matrix around point $(0, 0, 0)$ and unit axis $(u_x, u_y, u_z)$ with angle $theta$ (Rodrigues' Rotation Formula).



begin{bmatrix} cos theta +u_x^2 left(1-cos thetaright) & u_x u_y left(1-cos thetaright) - u_z sin theta & u_x u_z left(1-cos thetaright) + u_y sin theta \ u_y u_x left(1-cos thetaright) + u_z sin theta & cos theta + u_y^2left(1-cos thetaright) & u_y u_z left(1-cos thetaright) - u_x sin theta \ u_z u_x left(1-cos thetaright) - u_y sin theta & u_z u_y left(1-cos thetaright) + u_x sin theta & cos theta + u_z^2left(1-cos thetaright)
end{bmatrix}



How it is possible to generalize rotation matrix on $N$ dimension around zero point and $N-2$ dimensional unit axis with angle $theta$?






linear-algebra matrices rotations







share|cite|improve this question











$endgroup$




There is a 2d rotation matrix around point $(0, 0)$ with angle $theta$.



$$
left[ begin{array}{ccc}
cos(theta) & -sin(theta) \
sin(theta) & cos(theta) end{array} right]
$$



Next, there is a 3d rotation matrix around point $(0, 0, 0)$ and unit axis $(u_x, u_y, u_z)$ with angle $theta$ (Rodrigues' Rotation Formula).



begin{bmatrix} cos theta +u_x^2 left(1-cos thetaright) & u_x u_y left(1-cos thetaright) - u_z sin theta & u_x u_z left(1-cos thetaright) + u_y sin theta \ u_y u_x left(1-cos thetaright) + u_z sin theta & cos theta + u_y^2left(1-cos thetaright) & u_y u_z left(1-cos thetaright) - u_x sin theta \ u_z u_x left(1-cos thetaright) - u_y sin theta & u_z u_y left(1-cos thetaright) + u_x sin theta & cos theta + u_z^2left(1-cos thetaright)
end{bmatrix}



How it is possible to generalize rotation matrix on $N$ dimension around zero point and $N-2$ dimensional unit axis with angle $theta$?






linear-algebra matrices rotations




linear-algebra matrices rotations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Sep 17 '12 at 0:13









Sasha

61.1k5110182




61.1k5110182










asked Sep 16 '12 at 23:13









Ivan KochurkinIvan Kochurkin

5791720




5791720












  • $begingroup$
    @RobertIsrael could you please tell me the reference (name of the formula, article, book, etc.) for your formula above (a matrix that rotates the span of u and v by angle theta)? Thank you in advance.
    $endgroup$
    – user71237
    Apr 6 '13 at 8:13


















  • $begingroup$
    @RobertIsrael could you please tell me the reference (name of the formula, article, book, etc.) for your formula above (a matrix that rotates the span of u and v by angle theta)? Thank you in advance.
    $endgroup$
    – user71237
    Apr 6 '13 at 8:13
















$begingroup$
@RobertIsrael could you please tell me the reference (name of the formula, article, book, etc.) for your formula above (a matrix that rotates the span of u and v by angle theta)? Thank you in advance.
$endgroup$
– user71237
Apr 6 '13 at 8:13




$begingroup$
@RobertIsrael could you please tell me the reference (name of the formula, article, book, etc.) for your formula above (a matrix that rotates the span of u and v by angle theta)? Thank you in advance.
$endgroup$
– user71237
Apr 6 '13 at 8:13










1 Answer
1






active

oldest

votes


















16












$begingroup$

The definition is that $Ain M_{n}(mathbb{R})$ is called a rotation
matrix if there exist a unitary matrix $P$ s.t $P^{-1}AP$ is of
the form $$begin{pmatrix}cos(theta) &-sin(theta)\
sin(theta) & cos(theta)\
& & 1\
& & & 1\
& & & & 1\
& & & & & .\
& & & & & & .\
& & & & & & & .\
& & & & & & & & 1
end{pmatrix}$$



If we consider $A:mathbb{R}^{n}tomathbb{R}^{n}$ then the meaning
is that there exist an orthonormal basis where we rotate the $2-$dimensional
space spanned by the first two vectors by angle $theta$ and we fix
the other $n-2$ dimensions






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Lyx tip is appriciated for the matrices
    $endgroup$
    – Belgi
    Sep 16 '12 at 23:23










  • $begingroup$
    Ok, but is there efficient algorithm for finding every matrix item, based on special numbers, permutations or something else, exists?
    $endgroup$
    – Ivan Kochurkin
    Sep 16 '12 at 23:31






  • 13




    $begingroup$
    If $u$ and $v$ are two orthonormal vectors, a matrix that rotates the span of $u$ and $v$ by angle $theta$ is $A = I + sin(theta) ( v u^T - u v^T) + (cos(theta) - 1)( u u^T + v v^T)$. Thus $A_{ij} = delta_{ij} + sin(theta) (v_i u_j - u_i v_j) + (cos(theta)-1)(u_i u_j + v_i v_j)$.
    $endgroup$
    – Robert Israel
    Sep 16 '12 at 23:51






  • 1




    $begingroup$
    The $n-2$-dimensional space that is the orthogonal complement of my two vectors is fixed by the rotation, i.e. $Aw = w$ for $w$ in this space.
    $endgroup$
    – Robert Israel
    Sep 20 '12 at 7:44






  • 4




    $begingroup$
    @RobertIsrael can you point to a source of these equations?
    $endgroup$
    – gota
    Oct 9 '17 at 15:54












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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









16












$begingroup$

The definition is that $Ain M_{n}(mathbb{R})$ is called a rotation
matrix if there exist a unitary matrix $P$ s.t $P^{-1}AP$ is of
the form $$begin{pmatrix}cos(theta) &-sin(theta)\
sin(theta) & cos(theta)\
& & 1\
& & & 1\
& & & & 1\
& & & & & .\
& & & & & & .\
& & & & & & & .\
& & & & & & & & 1
end{pmatrix}$$



If we consider $A:mathbb{R}^{n}tomathbb{R}^{n}$ then the meaning
is that there exist an orthonormal basis where we rotate the $2-$dimensional
space spanned by the first two vectors by angle $theta$ and we fix
the other $n-2$ dimensions






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Lyx tip is appriciated for the matrices
    $endgroup$
    – Belgi
    Sep 16 '12 at 23:23










  • $begingroup$
    Ok, but is there efficient algorithm for finding every matrix item, based on special numbers, permutations or something else, exists?
    $endgroup$
    – Ivan Kochurkin
    Sep 16 '12 at 23:31






  • 13




    $begingroup$
    If $u$ and $v$ are two orthonormal vectors, a matrix that rotates the span of $u$ and $v$ by angle $theta$ is $A = I + sin(theta) ( v u^T - u v^T) + (cos(theta) - 1)( u u^T + v v^T)$. Thus $A_{ij} = delta_{ij} + sin(theta) (v_i u_j - u_i v_j) + (cos(theta)-1)(u_i u_j + v_i v_j)$.
    $endgroup$
    – Robert Israel
    Sep 16 '12 at 23:51






  • 1




    $begingroup$
    The $n-2$-dimensional space that is the orthogonal complement of my two vectors is fixed by the rotation, i.e. $Aw = w$ for $w$ in this space.
    $endgroup$
    – Robert Israel
    Sep 20 '12 at 7:44






  • 4




    $begingroup$
    @RobertIsrael can you point to a source of these equations?
    $endgroup$
    – gota
    Oct 9 '17 at 15:54
















16












$begingroup$

The definition is that $Ain M_{n}(mathbb{R})$ is called a rotation
matrix if there exist a unitary matrix $P$ s.t $P^{-1}AP$ is of
the form $$begin{pmatrix}cos(theta) &-sin(theta)\
sin(theta) & cos(theta)\
& & 1\
& & & 1\
& & & & 1\
& & & & & .\
& & & & & & .\
& & & & & & & .\
& & & & & & & & 1
end{pmatrix}$$



If we consider $A:mathbb{R}^{n}tomathbb{R}^{n}$ then the meaning
is that there exist an orthonormal basis where we rotate the $2-$dimensional
space spanned by the first two vectors by angle $theta$ and we fix
the other $n-2$ dimensions






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Lyx tip is appriciated for the matrices
    $endgroup$
    – Belgi
    Sep 16 '12 at 23:23










  • $begingroup$
    Ok, but is there efficient algorithm for finding every matrix item, based on special numbers, permutations or something else, exists?
    $endgroup$
    – Ivan Kochurkin
    Sep 16 '12 at 23:31






  • 13




    $begingroup$
    If $u$ and $v$ are two orthonormal vectors, a matrix that rotates the span of $u$ and $v$ by angle $theta$ is $A = I + sin(theta) ( v u^T - u v^T) + (cos(theta) - 1)( u u^T + v v^T)$. Thus $A_{ij} = delta_{ij} + sin(theta) (v_i u_j - u_i v_j) + (cos(theta)-1)(u_i u_j + v_i v_j)$.
    $endgroup$
    – Robert Israel
    Sep 16 '12 at 23:51






  • 1




    $begingroup$
    The $n-2$-dimensional space that is the orthogonal complement of my two vectors is fixed by the rotation, i.e. $Aw = w$ for $w$ in this space.
    $endgroup$
    – Robert Israel
    Sep 20 '12 at 7:44






  • 4




    $begingroup$
    @RobertIsrael can you point to a source of these equations?
    $endgroup$
    – gota
    Oct 9 '17 at 15:54














16












16








16





$begingroup$

The definition is that $Ain M_{n}(mathbb{R})$ is called a rotation
matrix if there exist a unitary matrix $P$ s.t $P^{-1}AP$ is of
the form $$begin{pmatrix}cos(theta) &-sin(theta)\
sin(theta) & cos(theta)\
& & 1\
& & & 1\
& & & & 1\
& & & & & .\
& & & & & & .\
& & & & & & & .\
& & & & & & & & 1
end{pmatrix}$$



If we consider $A:mathbb{R}^{n}tomathbb{R}^{n}$ then the meaning
is that there exist an orthonormal basis where we rotate the $2-$dimensional
space spanned by the first two vectors by angle $theta$ and we fix
the other $n-2$ dimensions






share|cite|improve this answer











$endgroup$



The definition is that $Ain M_{n}(mathbb{R})$ is called a rotation
matrix if there exist a unitary matrix $P$ s.t $P^{-1}AP$ is of
the form $$begin{pmatrix}cos(theta) &-sin(theta)\
sin(theta) & cos(theta)\
& & 1\
& & & 1\
& & & & 1\
& & & & & .\
& & & & & & .\
& & & & & & & .\
& & & & & & & & 1
end{pmatrix}$$



If we consider $A:mathbb{R}^{n}tomathbb{R}^{n}$ then the meaning
is that there exist an orthonormal basis where we rotate the $2-$dimensional
space spanned by the first two vectors by angle $theta$ and we fix
the other $n-2$ dimensions







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Mar 28 '16 at 12:45









Mark Amery

1286




1286










answered Sep 16 '12 at 23:23









BelgiBelgi

14.7k1156115




14.7k1156115












  • $begingroup$
    Lyx tip is appriciated for the matrices
    $endgroup$
    – Belgi
    Sep 16 '12 at 23:23










  • $begingroup$
    Ok, but is there efficient algorithm for finding every matrix item, based on special numbers, permutations or something else, exists?
    $endgroup$
    – Ivan Kochurkin
    Sep 16 '12 at 23:31






  • 13




    $begingroup$
    If $u$ and $v$ are two orthonormal vectors, a matrix that rotates the span of $u$ and $v$ by angle $theta$ is $A = I + sin(theta) ( v u^T - u v^T) + (cos(theta) - 1)( u u^T + v v^T)$. Thus $A_{ij} = delta_{ij} + sin(theta) (v_i u_j - u_i v_j) + (cos(theta)-1)(u_i u_j + v_i v_j)$.
    $endgroup$
    – Robert Israel
    Sep 16 '12 at 23:51






  • 1




    $begingroup$
    The $n-2$-dimensional space that is the orthogonal complement of my two vectors is fixed by the rotation, i.e. $Aw = w$ for $w$ in this space.
    $endgroup$
    – Robert Israel
    Sep 20 '12 at 7:44






  • 4




    $begingroup$
    @RobertIsrael can you point to a source of these equations?
    $endgroup$
    – gota
    Oct 9 '17 at 15:54


















  • $begingroup$
    Lyx tip is appriciated for the matrices
    $endgroup$
    – Belgi
    Sep 16 '12 at 23:23










  • $begingroup$
    Ok, but is there efficient algorithm for finding every matrix item, based on special numbers, permutations or something else, exists?
    $endgroup$
    – Ivan Kochurkin
    Sep 16 '12 at 23:31






  • 13




    $begingroup$
    If $u$ and $v$ are two orthonormal vectors, a matrix that rotates the span of $u$ and $v$ by angle $theta$ is $A = I + sin(theta) ( v u^T - u v^T) + (cos(theta) - 1)( u u^T + v v^T)$. Thus $A_{ij} = delta_{ij} + sin(theta) (v_i u_j - u_i v_j) + (cos(theta)-1)(u_i u_j + v_i v_j)$.
    $endgroup$
    – Robert Israel
    Sep 16 '12 at 23:51






  • 1




    $begingroup$
    The $n-2$-dimensional space that is the orthogonal complement of my two vectors is fixed by the rotation, i.e. $Aw = w$ for $w$ in this space.
    $endgroup$
    – Robert Israel
    Sep 20 '12 at 7:44






  • 4




    $begingroup$
    @RobertIsrael can you point to a source of these equations?
    $endgroup$
    – gota
    Oct 9 '17 at 15:54
















$begingroup$
Lyx tip is appriciated for the matrices
$endgroup$
– Belgi
Sep 16 '12 at 23:23




$begingroup$
Lyx tip is appriciated for the matrices
$endgroup$
– Belgi
Sep 16 '12 at 23:23












$begingroup$
Ok, but is there efficient algorithm for finding every matrix item, based on special numbers, permutations or something else, exists?
$endgroup$
– Ivan Kochurkin
Sep 16 '12 at 23:31




$begingroup$
Ok, but is there efficient algorithm for finding every matrix item, based on special numbers, permutations or something else, exists?
$endgroup$
– Ivan Kochurkin
Sep 16 '12 at 23:31




13




13




$begingroup$
If $u$ and $v$ are two orthonormal vectors, a matrix that rotates the span of $u$ and $v$ by angle $theta$ is $A = I + sin(theta) ( v u^T - u v^T) + (cos(theta) - 1)( u u^T + v v^T)$. Thus $A_{ij} = delta_{ij} + sin(theta) (v_i u_j - u_i v_j) + (cos(theta)-1)(u_i u_j + v_i v_j)$.
$endgroup$
– Robert Israel
Sep 16 '12 at 23:51




$begingroup$
If $u$ and $v$ are two orthonormal vectors, a matrix that rotates the span of $u$ and $v$ by angle $theta$ is $A = I + sin(theta) ( v u^T - u v^T) + (cos(theta) - 1)( u u^T + v v^T)$. Thus $A_{ij} = delta_{ij} + sin(theta) (v_i u_j - u_i v_j) + (cos(theta)-1)(u_i u_j + v_i v_j)$.
$endgroup$
– Robert Israel
Sep 16 '12 at 23:51




1




1




$begingroup$
The $n-2$-dimensional space that is the orthogonal complement of my two vectors is fixed by the rotation, i.e. $Aw = w$ for $w$ in this space.
$endgroup$
– Robert Israel
Sep 20 '12 at 7:44




$begingroup$
The $n-2$-dimensional space that is the orthogonal complement of my two vectors is fixed by the rotation, i.e. $Aw = w$ for $w$ in this space.
$endgroup$
– Robert Israel
Sep 20 '12 at 7:44




4




4




$begingroup$
@RobertIsrael can you point to a source of these equations?
$endgroup$
– gota
Oct 9 '17 at 15:54




$begingroup$
@RobertIsrael can you point to a source of these equations?
$endgroup$
– gota
Oct 9 '17 at 15:54


















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.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid
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