Find orthogonal matrices Let $A=\begin{bmatrix} 1 &   -1/2&-1/2  \\ -1/2  & 1&  -1/2\\ -1/2&-1/2 &1 \end{bmatrix}$. Is it possible to find explicitly orthogonal matrices $P, Q$ such that  $2A=PJP^t+QJQ^t$? Here $J$ is a matrix of all entries one. 

Note that $A$ has eigenvalues $0$, $3/2$, $3/2$, with null space spanned by 
$[1,1,1]^t$.  On the other hand, $J$ has eigenvalues $0,0,3$ with $[1,1,1]^t$ its eigenvector for $3$.  Consider an orthonormal basis 
$$u = \dfrac{1}{\sqrt{3}}\pmatrix{1\cr 1\cr 1\cr}, \ v = \dfrac{1}{\sqrt{2}} \pmatrix{1\cr -1\cr 0\cr},\ w = \dfrac{1}{\sqrt{6}} \pmatrix{1\cr 1\cr -2\cr}$$
We take $P$ so that $P v = u$, $P u = v$ and $P w = w$, and 
$Q$ so that $Qu = w$, $Qv = v$, $Qw = u$.
$$ P =  \dfrac{1}{6} \pmatrix{1 + 2\sqrt{6} & 1 & -2 + \sqrt{6} \cr
                              1 & 1 - 2 \sqrt{6}  & -2 - \sqrt{6}\cr
                              -2 + \sqrt{6}  &  -2 - \sqrt{6} & 4\cr},
    \ Q = \dfrac{1}{18} \pmatrix{2 \sqrt{18} + 9 & 2 \sqrt{18} - 9 & -\sqrt{18}\cr
2 \sqrt{18} - 9 & 2 \sqrt{18} + 9 & -\sqrt{18}\cr
-\sqrt{18} & -\sqrt{18} & -4 \sqrt{18}}$$
