Redefine the P matrix so that it’s made of eigenvectors with integer components. 
I’m trying to redefine the P matrix so that it’s made of eigenvectors with integer components (which will give the same diagonalizing matrix). But I’m stuck because I’m not sure how to redefine all the columns. I think the first column will be redefined as 1,-1,1,-1. But for the rest of the columns, I'm stuck. Please help me out, if possible. Thank you.
(Original Matrix -> H)

$$A = \begin{pmatrix}
 \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\
 \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\
 -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
 \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
\end{pmatrix}$$
$$A = P D P^{-1} =\begin{pmatrix}
 -1 & 1 & -1 & 1 \\
 1 & 0 & 0 & 1 \\
 -1 & 0 & 1 & 0 \\
 1 & 1 & 0 & 0 \\
\end{pmatrix} \begin{pmatrix}
 -1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1 \\
\end{pmatrix}\begin{pmatrix}
 -\frac{1}{4} & \frac{1}{4} & -\frac{1}{4} & \frac{1}{4} \\
 \frac{1}{4} & -\frac{1}{4} & \frac{1}{4} & \frac{3}{4} \\
 -\frac{1}{4} & \frac{1}{4} & \frac{3}{4} & \frac{1}{4} \\
 \frac{1}{4} & \frac{3}{4} & \frac{1}{4} & -\frac{1}{4} \\
\end{pmatrix}$$
Note: You can also factor our $\frac{1}{2}$ from $A$. This will change $D$ to just be two times what is shown, but $P$ and $P^{-1}$ will be the same.
