Let the square symmetric matrix $L\in\mathbb{R}^{n\times n}$ be positive semi-definite with vector $1_n$ spanning its null-space (i.e., vector $1_n$ is the eigenvector of $L$ corresponding to the eigenvalue $0$). Consider $L$ to have positive diagonal entries, while the negative entries appear only in off-diagonal positions.
A matrix $X\in\mathbb{n\times k}$, $k
I suppose that $X^T1_n=0_m$ needs be satisfied, but I'm not sure if this is sufficient (and correct).
Note that the positive definiteness of a symmetric matrix $A$ is equivalent to exclusively positive spectrum of $A$.