# A problem with definitions of rotation/reflection matrix/operator

I am a math undergraduate student taking a course called "Geometry and symmetry" and I have something I don't understand with the definition the lecture gave in class.

Definition: $T\,:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is called a linear rotation operator if there exist an orthonormal basis $B$ s.t $$\begin{pmatrix}cos(\theta) & -sin(\theta)\\ sin(\theta) & cos(\theta)\\ & & 1\\ & & & 1\\ & & & & 1\\ & & & & & & .\\ & & & & & & & .\\ & & & & & & & & .\\ & & & & & & & & 1\\ & & & & & & & & & 1 \end{pmatrix}$$.

Definition: $T\,:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is called a linear reflection operator if there exist an orthonormal basis $B$ s.t $$\begin{pmatrix}1\\ & 1\\ & & .\\ & & & .\\ & & & & .\\ & & & & & 1\\ & & & & & & -1 \end{pmatrix}$$.

The definitions for matrix $P$ are that a matrix is called linear reflection matrix if there is an orthogonal matrix $P$ s.t $P^{-1}AP$ is in one of the above forms, accordingly.

Later on, there is a theorem that says that is $T\,:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is orthogonal then there exist an orthonormal basis $B$ s.t $$[T]_{B}=\begin{pmatrix}R_{\theta_{1}}\\ & R_{\theta_{2}}\\ & & .\\ & & & .\\ & & & & .\\ & & & & & R_{\theta_{k}}\\ & & & & & & 1\\ & & & & & & & 1\\ & & & & & & & & .\\ & & & & & & & & & 1\\ & & & & & & & & & & -1\\ & & & & & & & & & & & .\\ & & & & & & & & & & & & .\\ & & & & & & & & & & & & & -1 \end{pmatrix};R_{\theta}:=\begin{pmatrix}cos(\theta) & -sin(\theta)\\ sin(\theta) & cos(\theta) \end{pmatrix}$$

My question is: if $T$ is orthogonal then if $A$ is a matrix s.t $Tv=Av$ then $A$ is orthogonal hence $|A|=1$ or $|A|=-1$. in the first case I learned that by $A$ is a linear rotation matrix, in the second case I learned that $A$ is linear reflection operator, so I deduce that there is an orthonormal basis in which $A$ have the form of the two matrix above - i.e rotating around some $2$ dimensional space fixing the other $n-2$ subspace of $\mathbb{R}^{n}$ or rotating around some vector fixing the other $n-1$ subspace of $\mathbb{R}^{n}$.

It seems that the matrix in the theorem is very different than the first two matrix in the question since it has $k$ (that can be $>1$) rotations, and it can reflects around more then $1$ vector...

Can someone please explain whats going on here? I expected the theorem to be the $[T]_{B}$ is one of the first two matrices in the question, instead I have in front of me a strange looking matrix that I can not classify as a rotation or a reflection matrix, so where in my reasoning I am wrong (I determined that $[T]_{B}$ is the first matrix in the question if $A$ above have det $1$ and the second if $A$ have det -1) ?

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You can use \ddots to get three dots in a diagonal, like so: $\ddots$ You should also use \cos and \sin to write $\cos\theta$ instead of $cos\theta$. – Rahul Jul 13 '12 at 23:25