Let $A = \left[\begin{array}{ccc} 1 &8 &-4\cr 8 &1 &4\cr -4 &4 &7 \end{array}\right]$. Find an orthogonal matrix $P$ with rational entries and a diagonal matrix $D$ such that $D = P^TAP$.
$P=$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$, $D=$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$