Let $A$ and $B$ be two full column rank real matrices of dimension $n \times m$, where $n \ge m$. Let $P$ be an $m\times m$ positive definite matrix.

Question: Does there always exist a symmetric $n \times n$ matrix $X$ such that the following holds?

 

$$\mathrm{tr}(P(A^\top XB+B^\top X A)) \ne 0$$

Let $A$ and $B$ be two full column rank real matrices of dimension $n \times m$, where $n \ge m$. Let $P$ be an $m\times m$ positive definite matrix.

Question: Does there always exist a symmetric $n \times n$ matrix $X$ such that the following holds?

$$\mathrm{tr}(P(A^\top XB+B^\top X A)) \ne 0$$

Let $A$ and $B$ be two full column-rank real matrices of dimension $n\times m$, $n\ge m$. Let $P$ be an $m\times m$ positive definite matrix.

Question. Does there always exist a symmetric $m\times m$ matrix $X$ such that $$ \mathrm{tr}(P(A^\top XB+B^\top X A))\ne 0\ \ ? $$

Let $A$ and $B$ be two full column rank real matrices of dimension $n \times m$, where $n \ge m$. Let $P$ be an $m\times m$ positive definite matrix.

Question: Does there always exist a symmetric $n \times n$ matrix $X$ such that the following holds?

$$\mathrm{tr}(P(A^\top XB+B^\top X A)) \ne 0$$