From complex rotation matrix to real matrix
Let's consider $R_{n}$ to be an $n \times n$ real rotation matrix. $R_{n}$ can be diagonalized with a unitary matrix $U_{n}$ to $D_{n}$, which is composed of blocs of $ \left( \begin{array}{ccc} e^{j\theta} & 0 \\ 0 & e^{-j\theta} \end{array} \right) $ and eventually ones.
But what I want is to have a real rotation matrix decomposition, with blocs of $ \left( \begin{array}{ccc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right)$ and ones.
How do I get the change of basis matrix that allows me to get such a decomposition ? Can I derive it from the $U_{n}$ matrix and the eigenvalues (the $\theta$s) ?
$\endgroup$ 11 Answer
$\begingroup$Notice that $$\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix} = \begin{pmatrix}\tfrac{1}{\sqrt{2}} & -\tfrac{j}{\sqrt{2}} \\ -\tfrac{j}{\sqrt{2}} & \tfrac{1}{\sqrt{2}}\end{pmatrix} \begin{pmatrix}e^{j\theta} & 0\\ 0 & e^{-j\theta}\end{pmatrix} \begin{pmatrix}\tfrac{1}{\sqrt{2}} & \tfrac{j}{\sqrt{2}} \\ \tfrac{j}{\sqrt{2}} & \tfrac{1}{\sqrt{2}}\end{pmatrix}.$$
So we can define $W_n$ to be a block diagonal matrix with the same block structure as $D_n$ such that for each $\begin{pmatrix}e^{j\theta} & 0\\ 0 & e^{-j\theta}\end{pmatrix}$ block in $D_n$, the corresponding block in $W_n$ is $\begin{pmatrix}\tfrac{1}{\sqrt{2}} & \tfrac{j}{\sqrt{2}} \\ \tfrac{j}{\sqrt{2}} & \tfrac{1}{\sqrt{2}}\end{pmatrix}$, and for each $\begin{pmatrix}1\end{pmatrix}$ block in $D_n$, the corresponding block in $W_n$ is also $\begin{pmatrix}1\end{pmatrix}$.
Then, $W_n^*D_nW_n = W_n^*U_n^*R_nU_nW_n$ will be block diagonal with several blocks of the form $\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$ and then several ones.
$\endgroup$ 1More in general
"Zoraya ter Beek, age 29, just died by assisted suicide in the Netherlands. She was physically healthy, but psychologically depressed. It's an abomination that an entire society would actively facilitate, even encourage, someone ending their own life because they had no hope. Th…"
