When using EM algorithm for estimating the state matrices ($\mathbf{A}$, $\mathbf{C}$) of a linear state-space model (as shown below), \begin{align*} x_k &= \mathbf{A} x_{k-1} + w_k\\ y_k &= \mathbf{C} x_{k} + v_k \end{align*} I found that the state matrix $\mathbf{A}$ loses its parameterization. What I mean by that is, I start with an initial estimate of the form $\mathbf{A}^{(0)} = \begin{bmatrix} 0 & I \\ E(\theta) & F(\theta) \end{bmatrix}$ but I lose this form of $\mathbf{A}$ after a couple iterations of EM. My question is: given a fully populated EM estimate of $\hat{\mathbf{A}} = \begin{bmatrix} \hat{A}_{11} & \hat{A}_{12} \\ \hat{A}_{21} & \hat{A}_{22} \end{bmatrix}$, is it possible to rotate $\hat{\mathbf{A}}$ to atleast have the structure of $\begin{bmatrix} 0 & I \\ \hat{E} & \hat{F} \end{bmatrix}$ ?
Note: I don't have the knowledge of $\hat{E}$ and $\hat{F}$. All I know is $\hat{\mathbf{A}}$ and the form of $[0 \;\;I]$ of the top block of parameterized $\mathbf{A}$