I am trying to prove the existence of a solution to the matrix equation $$X\cdot M\cdot X'=0$$ where $X$ is an unknown matrix of dimensions $n$ x $m$, and $M$ is a known squared matrix $m$ x $m$. This, of course, has the zero as a solution, but the problem has the restrictions that $X$ is block-diagonal, each block with the same number of columns, say $l$, and there is a known partition of the indexes of such columns such that in each block there is an identity matrix in the columns given by the partition. Finally, using the same dimensions of blocks, we know that $M$ has a zero block diagonal.
For example: $$\left[\begin{array}{c c c c c c}1&x_1&x_2&0&0&0\\ 0&0&0&x_3&1&0\\0&0&0&x_4&0&1\end{array}\right]\cdot\left[\begin{array}{c c}0&M_1\\M_2&0\end{array}\right]\cdot\left[\begin{array}{c c c c c c}1&x_1&x_2&0&0&0\\ 0&0&0&x_3&1&0\\0&0&0&x_4&0&1\end{array}\right]^\prime=0$$ Here $l=3$ and the partition is ${1,\{2,3\}}$. So $M_1$ and $M2$ have the appropriate dimension. If necessary the $M_i$ matrices are non-zero, real and symmetric, but importantly, they can have a zero determinant.
Is there a way to solve this numerically? I've tried using fsolve in Matlab, but it does not behave well for some examples. maybe there is an alternative method that would allow a more direct way to impose the restrictions in the problem.