# fixed point problem for orthogonal transformation

Let T : $\mathbb{R^3} \rightarrow \mathbb{R^3}$ be an orthogonal transformation such that det $T = 1$ and T is not the identity linear transformation. Let S $\mathbb{R^3}$ be the unit sphere, i.e., S = {$(x; y; z):x^2 + y^2 + z^2 = 1$}: Show that T fixes exactly two points on S.

can anyone help me to solve this problem.thanks.

Intuitively, these should be the points for which the axis or the rotation intersects the sphere. To find these for an explicit given $T$ we need to solve $T(v)=v$ since this geometrically characterizes the axis. I suppose you need to think about how to argue that $\lambda = 1$ is in fact the only real eigenvalue of $T$. Hint: the information that $T$ is not the identity is important to this goal. Also, for an orthogonal transformation the matrix $[T]=A$ will have $A^TA=I$. Take the determinant and recall the determinant is equal to the product of the eigenvalues (possibly complex). Think about this and you can construct the answer easily.
• I assume you know that $det(AB)=det(A)det(B)$. Commented Sep 23, 2012 at 2:10