Suppose I have the following matrix constructed from some orthogonal matrix $O$ and a $\pm 1$ diagonal matrix $D=diag(\pm1,\dots,\pm1)$ $$ A = O D O^{-1} D. $$ Is there a simple way to evaluate $A^n$ for positive integer $n$ in a similar way to e.g. a diagonalizable matrix $B$? $$ B^n = (P L P^{-1})^n = P L^n P^{-1} . $$ I feel this ought to be the case given how $A$ is decomposed into a product of orthogonal and diagonal matrices but alas it is not clear to me if this is always so. Of course $A$ is also an orthogonal matrix and can itself be diagonalized $$ A = O D O^{-1} D = RTR^{-1} $$ but do $R,T$ have clear expressions in terms of $O,D$?
EDIT: Some further work on my end indicates that the same question when $$ A = D^{1/2} O D O^{-1} D^{1/2}, $$ is also acceptable. Answers in this direction would be helpful, though the original question is still my primary goal.