Rotation operator is positive.











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I want to know if my proof is right.
the problem is:



Let V be $mathbb{R^2}$, with the standard inner product. If $theta$ is a real number, let T
be the linear operator 'rotation through $theta$,
$T_{theta}(x_1, x_2) = (x_l cos theta - x_2 sin theta, x_1 sin theta + x_2 cos theta).$
For which values of $theta$ is $T_{theta}$ a positive operator?



The conditions for $T_theta$ to be a positive operator are $T=T^*$ and $langle Talpha,alpharangle >0$.



First of all, I computed the associated matrix $[T]_b=begin{bmatrix} costheta & -sintheta \ sintheta & costheta
end{bmatrix}$
.



The only way that $T=T^*$ is $sintheta=0$ or $theta=pi k$ with $kin mathbb{Z}$.



Then, $langle Talpha,alpharangle=costheta(x_1^2+x_2^2)$, to $langle Talpha,alpharangle$ be positive $theta$ must be in the first or fourth quadrant.



Hence, $theta=2pi k$ with $kin mathbb{Z}$.



Is anything wrong?










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  • 2




    That's perfect. In other words, only the identity is positive among the rotations.
    – Berci
    Nov 17 at 18:47










  • Nice, thank you I didn't realize that!!
    – Bayesian guy
    Nov 18 at 3:47















up vote
0
down vote

favorite












I want to know if my proof is right.
the problem is:



Let V be $mathbb{R^2}$, with the standard inner product. If $theta$ is a real number, let T
be the linear operator 'rotation through $theta$,
$T_{theta}(x_1, x_2) = (x_l cos theta - x_2 sin theta, x_1 sin theta + x_2 cos theta).$
For which values of $theta$ is $T_{theta}$ a positive operator?



The conditions for $T_theta$ to be a positive operator are $T=T^*$ and $langle Talpha,alpharangle >0$.



First of all, I computed the associated matrix $[T]_b=begin{bmatrix} costheta & -sintheta \ sintheta & costheta
end{bmatrix}$
.



The only way that $T=T^*$ is $sintheta=0$ or $theta=pi k$ with $kin mathbb{Z}$.



Then, $langle Talpha,alpharangle=costheta(x_1^2+x_2^2)$, to $langle Talpha,alpharangle$ be positive $theta$ must be in the first or fourth quadrant.



Hence, $theta=2pi k$ with $kin mathbb{Z}$.



Is anything wrong?










share|cite|improve this question




















  • 2




    That's perfect. In other words, only the identity is positive among the rotations.
    – Berci
    Nov 17 at 18:47










  • Nice, thank you I didn't realize that!!
    – Bayesian guy
    Nov 18 at 3:47













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I want to know if my proof is right.
the problem is:



Let V be $mathbb{R^2}$, with the standard inner product. If $theta$ is a real number, let T
be the linear operator 'rotation through $theta$,
$T_{theta}(x_1, x_2) = (x_l cos theta - x_2 sin theta, x_1 sin theta + x_2 cos theta).$
For which values of $theta$ is $T_{theta}$ a positive operator?



The conditions for $T_theta$ to be a positive operator are $T=T^*$ and $langle Talpha,alpharangle >0$.



First of all, I computed the associated matrix $[T]_b=begin{bmatrix} costheta & -sintheta \ sintheta & costheta
end{bmatrix}$
.



The only way that $T=T^*$ is $sintheta=0$ or $theta=pi k$ with $kin mathbb{Z}$.



Then, $langle Talpha,alpharangle=costheta(x_1^2+x_2^2)$, to $langle Talpha,alpharangle$ be positive $theta$ must be in the first or fourth quadrant.



Hence, $theta=2pi k$ with $kin mathbb{Z}$.



Is anything wrong?










share|cite|improve this question















I want to know if my proof is right.
the problem is:



Let V be $mathbb{R^2}$, with the standard inner product. If $theta$ is a real number, let T
be the linear operator 'rotation through $theta$,
$T_{theta}(x_1, x_2) = (x_l cos theta - x_2 sin theta, x_1 sin theta + x_2 cos theta).$
For which values of $theta$ is $T_{theta}$ a positive operator?



The conditions for $T_theta$ to be a positive operator are $T=T^*$ and $langle Talpha,alpharangle >0$.



First of all, I computed the associated matrix $[T]_b=begin{bmatrix} costheta & -sintheta \ sintheta & costheta
end{bmatrix}$
.



The only way that $T=T^*$ is $sintheta=0$ or $theta=pi k$ with $kin mathbb{Z}$.



Then, $langle Talpha,alpharangle=costheta(x_1^2+x_2^2)$, to $langle Talpha,alpharangle$ be positive $theta$ must be in the first or fourth quadrant.



Hence, $theta=2pi k$ with $kin mathbb{Z}$.



Is anything wrong?







linear-algebra






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 18 at 2:44









Paul Sinclair

18.9k21440




18.9k21440










asked Nov 17 at 18:22









Bayesian guy

4710




4710








  • 2




    That's perfect. In other words, only the identity is positive among the rotations.
    – Berci
    Nov 17 at 18:47










  • Nice, thank you I didn't realize that!!
    – Bayesian guy
    Nov 18 at 3:47














  • 2




    That's perfect. In other words, only the identity is positive among the rotations.
    – Berci
    Nov 17 at 18:47










  • Nice, thank you I didn't realize that!!
    – Bayesian guy
    Nov 18 at 3:47








2




2




That's perfect. In other words, only the identity is positive among the rotations.
– Berci
Nov 17 at 18:47




That's perfect. In other words, only the identity is positive among the rotations.
– Berci
Nov 17 at 18:47












Nice, thank you I didn't realize that!!
– Bayesian guy
Nov 18 at 3:47




Nice, thank you I didn't realize that!!
– Bayesian guy
Nov 18 at 3:47















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