$AX^\top+XA^\top=0$

Question

If $A\in\mathbb R^{N\times p}$, for $p<N$, is there anything simple I can say about the solutions $X\in \mathbb R^{N\times p}$ of the equation $AX^\top+XA^\top=0$ ?

Answer 1

When dealing with matrix equations, a powerful tool is the conversion to a equivalent linear problem (in a bigger vector space) via the Kronecker product, as explained in Chapter 4 of Topics in matrix analysis by Horn and Johnson. As inferred from Problem 4.3.3 of the same book, the solutions of the system $AX^T+XA^T=0$ are the transposes of the solutions of $$\left(I\otimes A^T + (A^T\otimes I)P(N,N)\right)\cdot\text{vec}(X)=0,$$ where $P(N,N)=(E_{ij}^T)_{1\leq i,j\leq N}$ and $E_{ij}$ are the elementary matrices.

Hence, the space of solutions is wholly determined by the matrix $$I\otimes A^T + (A^T\otimes I)P(N,N).$$