I have a matrix inequality problem (don't know if this holds):
$A \in \mathbb{R}^{m \times n}$, $B_i \in \mathbb{R}^{n \times n}$, $i=1, 2, \dots, m$. $B_i$ is Hermitian. $Ax>0$ doesn't have a solution. $Ax=0$ has non-zero solutions. There exists a non-zero vector $\lambda \in \mathbb{R}^m$ such that $\lambda^TA=0$. And $$ Ax+\left[ \begin{aligned} x^TB_1x \\ x^TB_2x \\ \cdots \\ x^TB_mx \end{aligned} \right] > 0 $$ doesn't have a solution. Prove that $\left( \sum\limits_{i=1}^{m}{\lambda_i B_i} \right)X = A^T$ has a solution.
I have learned some knowledge about matrix but I didn't find any useful theorems.
Can anyone give me some ideas or theorems that may help?