How to check diagonalization of orthogonal matrix?

Question

Is an orthogonal matrix diagonalizable over $\mathbb{R}$ or $\mathbb{C}$ ?

I know that if all eigenvalues of a matrix are distinct then it is diagonalizable. G.m. of each eigenvalue = its A.m. And so on but sometimes there are complex matrices and I'm unable to find eigenvalues. Please help me.

Answer 1

The matrix $\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$ is orthogonal, but it is not diagonalizable over $\mathbb R$ (it has no real eigenvalues). However, it is diagonalizable over $\mathbb C$. In fact, every real orthogonal matrix is diagonalizable over $\mathbb C$; this follows from the spectral theorem.

Answer 2

Also, every real orthogonal matrix over $\mathbb C$ is also unitarly diagonalizable(because, it'll be normal hence spectral decomposition).

Answer 3

Actually, if we consider rotation matrix of order $2\times 2$, then we know rotation matrix is orthogonal matrix.then rotation matrix have no real eigen value in $\mathbb{R}$, unless the rotation angle $A$ is either $0$ or $\pi$. That is rotation matrix have no eigenvector. Again $A$ linear operator on a real vector space need not have an eigenvector. That is, every orthogonal matrix is not always diagonalizable over $\mathbb{R}$, but it is always diagonalizable over $\mathbb{C}$,because every linear operator on a complex vector space has at least one eigenvector.