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could any one just give hints for the following?

$T\neq I$ is a orthogonal operator on $\mathbb{R}^3$ with $det T=1$, we need to show that $T$ fixes exactly $2$ points on $S^2$

well, I was just thinking by contradiction if it fixes $3$ points say $(x_1,x_2,x_3),(x_4,x_5,x_6),(x_7,x_8,x_9)\in S^2$ then calculated the matrix of $T$...am I going in right path?

Myshkin
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  • Have you determined the possible eigenvalues of $T$? Observe that a fixed point must belong to the eigenspace $\lambda=1$. If there are more than two, then the dimension of the eigenspace must be ... See my answer to another question for a related argument (and other answers there for more discussion). Undoubtedly the same argument has been given many times in this site. – Jyrki Lahtonen Jan 21 '13 at 08:34
  • every rotation has axis so it must have $1$ as an eigen value, I dont know more. – Myshkin Jan 21 '13 at 08:38
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    Good. How many fixed points will one axis give you? – Jyrki Lahtonen Jan 21 '13 at 08:39
  • Correct. What can you say about the dimension of the eigenspace of $\lambda=1$, if there are at least three fixed points? – Jyrki Lahtonen Jan 21 '13 at 08:41
  • $2$ , as eigen vectors generates a plane. – Myshkin Jan 21 '13 at 08:46
  • What's the product of all three eigenvalues? If you know two of them, then what about the third? – Jyrki Lahtonen Jan 21 '13 at 08:48
  • product of all eigen values will be $1$, the third must be $1$ – Myshkin Jan 21 '13 at 08:50
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    So at that point you have an orthogonal transformation $T$ on $\mathbb{R}^3$ with $\lambda=1$ as a multiplicity 3 eigenvalue. That leaves very few options for $T$. Or another way to make further progress: a 2-dimensional eigenspace will intersect the orthogonal complement of the axis of rotation (=the plane of rotation) in a non-trivial way. If a rotation (on that plane) fixes a non-zero point, then ... – Jyrki Lahtonen Jan 21 '13 at 08:55
  • $T$ must be identity transformation. i did not understand the 2nd way, I am poor in english – Myshkin Jan 21 '13 at 09:03
  • Correct. You might benefit from typing up the full argument as an answer. That way you get more feedback. – Jyrki Lahtonen Jan 21 '13 at 09:04

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You are going the right path. Since you only want a hint: Compute the matrix with respect to a specific basis. Then use that the matrix is special orthogonal.

Julian Kuelshammer
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