We study the joint recovery of an unknown system matrix $A\in\mathbb R^{n\times n}$ and a rank-$r$ initial-state matrix $X_0\in\mathbb R^{n\times m}$ from partial observations of the state matrices $A^tX_0$, for $t=0,\ldots,T$. Each trajectory is observed at a fixed set of state coordinates, which may differ across trajectories. We derive necessary conditions for identifiability, including obstructions arising from incomplete spatial coverage and insufficient dynamical span, as well as lower bounds on the number of observations. We then characterize the Jacobian of the measurement map, account for the intrinsic symmetry of the low-rank factorization, and establish a rank-certificate principle yielding generic local identifiability. When the trajectories whose initial states form a basis of $\operatorname{range}(X_0)$ are fully observed, we obtain necessary and sufficient rank conditions for global recovery, together with explicit reconstruction formulas. We formulate the joint recovery problem as a nonlinear least-squares problem over the system matrix $A$ and the low-rank factors of $X_0$, and solve it using Adam. Numerical experiments on synthetic and real-world data evaluate the recovery performance under different spatial sampling rates and observation horizons.

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