Question about rotation matrices
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How do I prove that, if for a $2 times 2$ matrix $A$ and a fixed integer $n> 0$ we have that $$A^n= begin{bmatrix} cos{x}& -sin{x}\ sin{x} & cos{x}end{bmatrix} ,$$
for some real $x$, then $A$ is also a rotation matrix? I know that it is very obvious, especially if you think about the isomorphism with complex numbers, but I can't seem to come up with a simple and rigorous proof.
linear-algebra rotations
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up vote
4
down vote
favorite
How do I prove that, if for a $2 times 2$ matrix $A$ and a fixed integer $n> 0$ we have that $$A^n= begin{bmatrix} cos{x}& -sin{x}\ sin{x} & cos{x}end{bmatrix} ,$$
for some real $x$, then $A$ is also a rotation matrix? I know that it is very obvious, especially if you think about the isomorphism with complex numbers, but I can't seem to come up with a simple and rigorous proof.
linear-algebra rotations
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
How do I prove that, if for a $2 times 2$ matrix $A$ and a fixed integer $n> 0$ we have that $$A^n= begin{bmatrix} cos{x}& -sin{x}\ sin{x} & cos{x}end{bmatrix} ,$$
for some real $x$, then $A$ is also a rotation matrix? I know that it is very obvious, especially if you think about the isomorphism with complex numbers, but I can't seem to come up with a simple and rigorous proof.
linear-algebra rotations
How do I prove that, if for a $2 times 2$ matrix $A$ and a fixed integer $n> 0$ we have that $$A^n= begin{bmatrix} cos{x}& -sin{x}\ sin{x} & cos{x}end{bmatrix} ,$$
for some real $x$, then $A$ is also a rotation matrix? I know that it is very obvious, especially if you think about the isomorphism with complex numbers, but I can't seem to come up with a simple and rigorous proof.
linear-algebra rotations
linear-algebra rotations
edited 6 hours ago
GNUSupporter 8964民主女神 地下教會
12.8k72445
12.8k72445
asked 6 hours ago
Tanny Sieben
30117
30117
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3 Answers
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up vote
4
down vote
"Very obvious" or not, it's not true. If $A=begin{bmatrix}0&1\1&0end{bmatrix}$ then $A^2=I$ but $A$ is not a rotation.
I see. So this means that the nth roots of a 2x2 rotation matrix are not necessarily rotation matrices? How would you then determine the nth roots of a rotation matrix?
– Tanny Sieben
6 hours ago
2
I doubt there's a simple algorithm to find all the $n$th roots. For example $I$ has infinitely many square roots, only two of which are rotations. Otoh if you just want $n$ $n$th roots that's easy. Say $A_theta$ is the rotation through the angle $theta$; then $B^n=A_theta$ if $B=A_{(theta+2pi k)/n}$, $k=1,dots,n$.
– David C. Ullrich
5 hours ago
I figured that too; What about if $A^n = R$ where $R$ is a 2x2 rotation matrix different from the identity? Then would A necessarily be a rotation matrix?
– Tanny Sieben
5 hours ago
add a comment |
up vote
2
down vote
Even for the case $A^2=R$ there are many possible roots.
Look for instance at:
https://www.maa.org/sites/default/files/pdf/cms_upload/Square_Roots-Sullivan13884.pdf
With the additional constraint $det(R)=1$ here you get $$operatorname{tr}(A)A=Rpm I$$
and have to discuss according values of $x$, whether the trace is zero or not and so on.
- e.g. $R=I$ and null trace, gives $begin{pmatrix}a&b\c&-aend{pmatrix}$ with $a^2+bc=1$
[ $a=0,b=1,c=1$ is the case presented by David C. Ulrich in his answer ]
For $A^n=R$ you get even more solutions.
Starting with $n=2p$ even, you have to solve $A^p=B$ for every $B^2=R$ with $B$ not necessarily a rotation already...
add a comment |
up vote
0
down vote
$DeclareMathOperator{Tr}{Tr}$For the case $A^2=R$ where $R$ is a 2x2 rotation different from identity, we can apply Sullivan's article quoted by zwim, and find:
If $R=begin{pmatrix}-1\&-1end{pmatrix}$, then $A=begin{pmatrix}alpha&beta\gamma&-alphaend{pmatrix}$ with $alpha^2+betagamma=-1$.
If $R$ is any other rotation over an angle $phi$, then $displaystyle A=pmfrac{R+I}{sqrt{2cosphi+2}}$. Working through the possibilities, it means that $A$ is a rotation matrix over either $frac 12phi$ or $frac 12phi + pi$.
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
"Very obvious" or not, it's not true. If $A=begin{bmatrix}0&1\1&0end{bmatrix}$ then $A^2=I$ but $A$ is not a rotation.
I see. So this means that the nth roots of a 2x2 rotation matrix are not necessarily rotation matrices? How would you then determine the nth roots of a rotation matrix?
– Tanny Sieben
6 hours ago
2
I doubt there's a simple algorithm to find all the $n$th roots. For example $I$ has infinitely many square roots, only two of which are rotations. Otoh if you just want $n$ $n$th roots that's easy. Say $A_theta$ is the rotation through the angle $theta$; then $B^n=A_theta$ if $B=A_{(theta+2pi k)/n}$, $k=1,dots,n$.
– David C. Ullrich
5 hours ago
I figured that too; What about if $A^n = R$ where $R$ is a 2x2 rotation matrix different from the identity? Then would A necessarily be a rotation matrix?
– Tanny Sieben
5 hours ago
add a comment |
up vote
4
down vote
"Very obvious" or not, it's not true. If $A=begin{bmatrix}0&1\1&0end{bmatrix}$ then $A^2=I$ but $A$ is not a rotation.
I see. So this means that the nth roots of a 2x2 rotation matrix are not necessarily rotation matrices? How would you then determine the nth roots of a rotation matrix?
– Tanny Sieben
6 hours ago
2
I doubt there's a simple algorithm to find all the $n$th roots. For example $I$ has infinitely many square roots, only two of which are rotations. Otoh if you just want $n$ $n$th roots that's easy. Say $A_theta$ is the rotation through the angle $theta$; then $B^n=A_theta$ if $B=A_{(theta+2pi k)/n}$, $k=1,dots,n$.
– David C. Ullrich
5 hours ago
I figured that too; What about if $A^n = R$ where $R$ is a 2x2 rotation matrix different from the identity? Then would A necessarily be a rotation matrix?
– Tanny Sieben
5 hours ago
add a comment |
up vote
4
down vote
up vote
4
down vote
"Very obvious" or not, it's not true. If $A=begin{bmatrix}0&1\1&0end{bmatrix}$ then $A^2=I$ but $A$ is not a rotation.
"Very obvious" or not, it's not true. If $A=begin{bmatrix}0&1\1&0end{bmatrix}$ then $A^2=I$ but $A$ is not a rotation.
answered 6 hours ago
David C. Ullrich
57.6k43891
57.6k43891
I see. So this means that the nth roots of a 2x2 rotation matrix are not necessarily rotation matrices? How would you then determine the nth roots of a rotation matrix?
– Tanny Sieben
6 hours ago
2
I doubt there's a simple algorithm to find all the $n$th roots. For example $I$ has infinitely many square roots, only two of which are rotations. Otoh if you just want $n$ $n$th roots that's easy. Say $A_theta$ is the rotation through the angle $theta$; then $B^n=A_theta$ if $B=A_{(theta+2pi k)/n}$, $k=1,dots,n$.
– David C. Ullrich
5 hours ago
I figured that too; What about if $A^n = R$ where $R$ is a 2x2 rotation matrix different from the identity? Then would A necessarily be a rotation matrix?
– Tanny Sieben
5 hours ago
add a comment |
I see. So this means that the nth roots of a 2x2 rotation matrix are not necessarily rotation matrices? How would you then determine the nth roots of a rotation matrix?
– Tanny Sieben
6 hours ago
2
I doubt there's a simple algorithm to find all the $n$th roots. For example $I$ has infinitely many square roots, only two of which are rotations. Otoh if you just want $n$ $n$th roots that's easy. Say $A_theta$ is the rotation through the angle $theta$; then $B^n=A_theta$ if $B=A_{(theta+2pi k)/n}$, $k=1,dots,n$.
– David C. Ullrich
5 hours ago
I figured that too; What about if $A^n = R$ where $R$ is a 2x2 rotation matrix different from the identity? Then would A necessarily be a rotation matrix?
– Tanny Sieben
5 hours ago
I see. So this means that the nth roots of a 2x2 rotation matrix are not necessarily rotation matrices? How would you then determine the nth roots of a rotation matrix?
– Tanny Sieben
6 hours ago
I see. So this means that the nth roots of a 2x2 rotation matrix are not necessarily rotation matrices? How would you then determine the nth roots of a rotation matrix?
– Tanny Sieben
6 hours ago
2
2
I doubt there's a simple algorithm to find all the $n$th roots. For example $I$ has infinitely many square roots, only two of which are rotations. Otoh if you just want $n$ $n$th roots that's easy. Say $A_theta$ is the rotation through the angle $theta$; then $B^n=A_theta$ if $B=A_{(theta+2pi k)/n}$, $k=1,dots,n$.
– David C. Ullrich
5 hours ago
I doubt there's a simple algorithm to find all the $n$th roots. For example $I$ has infinitely many square roots, only two of which are rotations. Otoh if you just want $n$ $n$th roots that's easy. Say $A_theta$ is the rotation through the angle $theta$; then $B^n=A_theta$ if $B=A_{(theta+2pi k)/n}$, $k=1,dots,n$.
– David C. Ullrich
5 hours ago
I figured that too; What about if $A^n = R$ where $R$ is a 2x2 rotation matrix different from the identity? Then would A necessarily be a rotation matrix?
– Tanny Sieben
5 hours ago
I figured that too; What about if $A^n = R$ where $R$ is a 2x2 rotation matrix different from the identity? Then would A necessarily be a rotation matrix?
– Tanny Sieben
5 hours ago
add a comment |
up vote
2
down vote
Even for the case $A^2=R$ there are many possible roots.
Look for instance at:
https://www.maa.org/sites/default/files/pdf/cms_upload/Square_Roots-Sullivan13884.pdf
With the additional constraint $det(R)=1$ here you get $$operatorname{tr}(A)A=Rpm I$$
and have to discuss according values of $x$, whether the trace is zero or not and so on.
- e.g. $R=I$ and null trace, gives $begin{pmatrix}a&b\c&-aend{pmatrix}$ with $a^2+bc=1$
[ $a=0,b=1,c=1$ is the case presented by David C. Ulrich in his answer ]
For $A^n=R$ you get even more solutions.
Starting with $n=2p$ even, you have to solve $A^p=B$ for every $B^2=R$ with $B$ not necessarily a rotation already...
add a comment |
up vote
2
down vote
Even for the case $A^2=R$ there are many possible roots.
Look for instance at:
https://www.maa.org/sites/default/files/pdf/cms_upload/Square_Roots-Sullivan13884.pdf
With the additional constraint $det(R)=1$ here you get $$operatorname{tr}(A)A=Rpm I$$
and have to discuss according values of $x$, whether the trace is zero or not and so on.
- e.g. $R=I$ and null trace, gives $begin{pmatrix}a&b\c&-aend{pmatrix}$ with $a^2+bc=1$
[ $a=0,b=1,c=1$ is the case presented by David C. Ulrich in his answer ]
For $A^n=R$ you get even more solutions.
Starting with $n=2p$ even, you have to solve $A^p=B$ for every $B^2=R$ with $B$ not necessarily a rotation already...
add a comment |
up vote
2
down vote
up vote
2
down vote
Even for the case $A^2=R$ there are many possible roots.
Look for instance at:
https://www.maa.org/sites/default/files/pdf/cms_upload/Square_Roots-Sullivan13884.pdf
With the additional constraint $det(R)=1$ here you get $$operatorname{tr}(A)A=Rpm I$$
and have to discuss according values of $x$, whether the trace is zero or not and so on.
- e.g. $R=I$ and null trace, gives $begin{pmatrix}a&b\c&-aend{pmatrix}$ with $a^2+bc=1$
[ $a=0,b=1,c=1$ is the case presented by David C. Ulrich in his answer ]
For $A^n=R$ you get even more solutions.
Starting with $n=2p$ even, you have to solve $A^p=B$ for every $B^2=R$ with $B$ not necessarily a rotation already...
Even for the case $A^2=R$ there are many possible roots.
Look for instance at:
https://www.maa.org/sites/default/files/pdf/cms_upload/Square_Roots-Sullivan13884.pdf
With the additional constraint $det(R)=1$ here you get $$operatorname{tr}(A)A=Rpm I$$
and have to discuss according values of $x$, whether the trace is zero or not and so on.
- e.g. $R=I$ and null trace, gives $begin{pmatrix}a&b\c&-aend{pmatrix}$ with $a^2+bc=1$
[ $a=0,b=1,c=1$ is the case presented by David C. Ulrich in his answer ]
For $A^n=R$ you get even more solutions.
Starting with $n=2p$ even, you have to solve $A^p=B$ for every $B^2=R$ with $B$ not necessarily a rotation already...
edited 5 hours ago
answered 5 hours ago
zwim
11.3k728
11.3k728
add a comment |
add a comment |
up vote
0
down vote
$DeclareMathOperator{Tr}{Tr}$For the case $A^2=R$ where $R$ is a 2x2 rotation different from identity, we can apply Sullivan's article quoted by zwim, and find:
If $R=begin{pmatrix}-1\&-1end{pmatrix}$, then $A=begin{pmatrix}alpha&beta\gamma&-alphaend{pmatrix}$ with $alpha^2+betagamma=-1$.
If $R$ is any other rotation over an angle $phi$, then $displaystyle A=pmfrac{R+I}{sqrt{2cosphi+2}}$. Working through the possibilities, it means that $A$ is a rotation matrix over either $frac 12phi$ or $frac 12phi + pi$.
add a comment |
up vote
0
down vote
$DeclareMathOperator{Tr}{Tr}$For the case $A^2=R$ where $R$ is a 2x2 rotation different from identity, we can apply Sullivan's article quoted by zwim, and find:
If $R=begin{pmatrix}-1\&-1end{pmatrix}$, then $A=begin{pmatrix}alpha&beta\gamma&-alphaend{pmatrix}$ with $alpha^2+betagamma=-1$.
If $R$ is any other rotation over an angle $phi$, then $displaystyle A=pmfrac{R+I}{sqrt{2cosphi+2}}$. Working through the possibilities, it means that $A$ is a rotation matrix over either $frac 12phi$ or $frac 12phi + pi$.
add a comment |
up vote
0
down vote
up vote
0
down vote
$DeclareMathOperator{Tr}{Tr}$For the case $A^2=R$ where $R$ is a 2x2 rotation different from identity, we can apply Sullivan's article quoted by zwim, and find:
If $R=begin{pmatrix}-1\&-1end{pmatrix}$, then $A=begin{pmatrix}alpha&beta\gamma&-alphaend{pmatrix}$ with $alpha^2+betagamma=-1$.
If $R$ is any other rotation over an angle $phi$, then $displaystyle A=pmfrac{R+I}{sqrt{2cosphi+2}}$. Working through the possibilities, it means that $A$ is a rotation matrix over either $frac 12phi$ or $frac 12phi + pi$.
$DeclareMathOperator{Tr}{Tr}$For the case $A^2=R$ where $R$ is a 2x2 rotation different from identity, we can apply Sullivan's article quoted by zwim, and find:
If $R=begin{pmatrix}-1\&-1end{pmatrix}$, then $A=begin{pmatrix}alpha&beta\gamma&-alphaend{pmatrix}$ with $alpha^2+betagamma=-1$.
If $R$ is any other rotation over an angle $phi$, then $displaystyle A=pmfrac{R+I}{sqrt{2cosphi+2}}$. Working through the possibilities, it means that $A$ is a rotation matrix over either $frac 12phi$ or $frac 12phi + pi$.
answered 28 mins ago
I like Serena
3,4521718
3,4521718
add a comment |
add a comment |
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