Let $A$ be a real $n\times n$ matrix. Suppose that there exists some positive integer $m$ such that $A^m=E$, the identity matrix. Show that $A$ is similar to an othogonal matrix; that is, there exists some invertible matrix $T$ such that $T^{-1}AT=B$ satisfies $B^TB=E$.
It is easy to see that $A$ is similar to the diagonal matrix in the complex field. However, what about the real field?
According to the last theorem on page 146 in ”A Classification of Matrices of Finite Order over C, R and Q”, the answer is yes: It is shown that a matrix of finite order over the real numbers is similar to a block diagonal matrix, where each block is either a 1x1 or 2x2 orthogonal matrix. Se the paper: Reginald Koo, ”A Classification of Matrices of Finite Order over C, R and Q”, Mathematics Magazine, Vol. 76, No. 2 (Apr., 2003), pp. 143−148 http://www.uam.es/personal_pdi/ciencias/otero/DOC/AI/3219311.pdf