Let $D=(d_{ij})_{i,j=1}^{n}$ be a diagonal matrix and $A=(a_{ij})_{i,j=1}^{n}$ a skew-symmetric matrix, where $d_{ij},a_{ij} \in \mathbb{R}$ for all $i,j$. Suppose the following equation holds:
$$
AA=D\,.
$$
In other words, $A$ is a square root of $D$.
Question: Does $d_{jj} \neq d_{kk}$ imply $a_{jk}=a_{kj}=0$?
Best Answer
Yes and this has nothing to do with skew-symmetric matrices. Since $D=A^2$ is a polynomial in $A$, we have $DA=AD$ and hence $d_{jj}a_{jk}=a_{jk}d_{kk}$. It follows that $a_{jk}=0$ when $d_{jj}\ne d_{kk}$.