Consider the matrices $$X = \begin{bmatrix} x_0 & x_1 & ... & x_{N-1} \end{bmatrix} \in \mathbb R^{n\times N}\\ U = \begin{bmatrix} u_0 & u_1 & ... & u_{N-1} \end{bmatrix} \in \mathbb R^{m\times N}\\ X_+ = \begin{bmatrix} x_1 & x_2 & ... & x_N \end{bmatrix}\in \mathbb R^{n\times N}.$$
The elements of the matrices are generated in the fashion of $$x_{i+1} = Ax_i + Bu_i, \quad i = 0,..., N-1, \quad x_0, U \text{ given.}$$ This can also be rewritten to $X_+ = AX+BU.$
Suppose $\begin{bmatrix} X \newline U\end{bmatrix}$ to be of full row rank $n+m$.
My question is, if this implies that $$\text{rank}(X_+-\lambda X) = n \quad \forall \lambda \in \mathbb C.$$
Edit: For a bit of context, the question I posed belongs to a data-driven control setting. Therefore we can assume that $x_0$ is not chosen arbitrary, but instead corresponds to the dynamics, i.e. $x_0 = Ax_{-1} + Bu_{-1}$, where steps $i\leq -1$ are those before we began to collect our matrices above. I think in this case, the implication does hold always.