I have a real $n\times n$ symmetric matrix of the form $(A)_{ij}=\varphi^{\vert i-j\vert}$, for example for $n=3$: $$ A=\begin{pmatrix} 1&\varphi&\varphi^2\\ \varphi&1&\varphi\\ \varphi^2&\varphi&1 \end{pmatrix}, \qquad\varphi\in\mathbb R. $$
Is there a simpler way, compared to a generic symmetric matrix, to find the orthogonal diagonalization $A=O^tDO$?
Extra question: do matrices of this form have a name?