When is the product of a skew-symmetric matrix $\mathbf{J}$ and a symmetric postive definite matrix $\mathbf{A}$ symmetric positive definite?

Question

Let $\mathbf{J}$ be a skew-symmetric matrix and $\mathbf{A}$ be a symmetric positive definite matrix. How can we construct $\mathbf{J}$ and $\mathbf{A}$ such that $\mathbf{J}\mathbf{A}$ is symmetric positive definite?

Answer 1

$$A=A^\top,\quad J=-J^\top$$ $$JA=(JA)^\top=A^\top J^\top=-AJ$$ $$AJ+JA=0$$

We can assume that $A$ is diagonal, with elements $a_i>0$. Then the $(i,j)$ element of $AB$ is $a_ib_{i,j}$, and the $(i,j)$ element of $BA$ is $b_{i,j}a_j$. So if the anticommutator $AB+BA$ vanishes, then

$$(a_i+a_j)b_{i,j}=0$$

which implies $B=0$.