Diagonal matrix equation

Question

Let $D$ be a 3-by-3 positive definite diagonal matrix, so all 3 diagonal entries are positive. Then is it possible to find all solutions $X$ which satisfy $$ X^T D X = D $$ Certainly $X = \pm I$ satisfy the equation, but how does one find all the nontrivial solutions?

Answer 1

$X^TDX=D$ means that $(D^{1/2}XD^{-1/2})^T(D^{1/2}XD^{-1/2})=D^{-1/2}X^TDXD^{-1/2}=I$. Therefore the general solution is given by $X=D^{-1/2}QD^{1/2}$ where $Q$ is any orthogonal matrix.