If $H$ is a projection matrix and $J$ is a matrix of ones, is it always true that $HJ = J$?

Question

I'm trying to prove that if $H$ is a projection matrix and $J$ is a matrix of ones, is it always true that $HJ = J$?

For example, $$ \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} $$

I'm not sure if this is true in general. Any guidance is appreciated.

Answer 1

The question is equivalent to asking whether $(1,1,\dots,1)$ is always an eigenvector of a projection matrix. This is clearly false, since that vector may not be preserved by the projection – take $\begin{bmatrix}1&0\\0&0\end{bmatrix}$ for example.

Answer 2

$ \def\o{{\large\tt1}} \def\L{\left}\def\R{\right}\def\LR#1{\L(#1\R)} \def\m#1{\left[\begin{array}{r}#1\end{array}\right]} $A small random counter-example for which $\,H=H^2\;$ but $\;HJ\ne J$ $$\eqalign{ H &= \frac{1}{693}\m{ 404 & 34 & 340 \\ 34 & 689 & -40 \\ 340 & -40 & 293 \\ } \qquad HJ &= \frac{1}{693}\m{ 778 & 778 & 778 \\ 683 & 683 & 683 \\ 593 & 593 & 593 \\ } \\\\ }$$ Note, however, that $\,HJ = \LR{H\o}\o^T = h\o^T$
So the product in question is always a rank-$\tt1$ matrix with constant rows.