Theorem (Rank condition for controllability)

Let $A\in M_{n}(\mathbb{R}),B\in M_{n\times m}(\mathbb{R})$. Then, $$\text{Image}(W(t_{0},t_{1}))=\text{Image}(W_{T})=\text{Image}(\mathcal{C}_{A,B})$$where $$\mathcal{C}_{A,B}=[B,AB,A^{2}B, \dots,A^{n-1}B]$$ is called the Controllability Matrix for the pair $(A,B)$. We then say an LTIC system with the pair $(A,B)$ is controllable for any pair $(t_{0},t_{1})$ if and only if $$\text{rank}(\mathcal{C}_{A,B})=n$$or equivalently: $$\text{Image}(\mathcal{C}_{A,B})=\mathbb{R}^{n}$$

Theorem (Rank Condition for Observability)

Consider an LTIC system system: $$\begin{cases} \dot{x}(t)=Ax(t)+Bu(t) \\ y(t)=Cx(t)+Du(t) \end{cases}$$then we have that $$\begin{align*} \text{Image}(M(t_{0},t_{1}))=\text{Image}(M_{T})\\ \text{Ker}(M(t_{0},t_{1}))=\text{Ker}(M_{T}) \end{align*}$$ where $$M_{T}=\begin{bmatrix}C \\ CA \\ \ddots \\ CA^{n-1} \end{bmatrix}^{\top} \begin{bmatrix}C \\ CA \\ \ddots \\ CA^{n-1} \end{bmatrix}$$Therefore $\text{Image}(M(t_{0},t_{1}))$ and $\text{Ker}(M(t_{0},t_{1}))$ are independent of $t_{0}$ and $t_{1}$. Moreover, $$\text{Image}(M(t_{0},t_{1}))=\text{Image}(M_{T})=\text{Image}(\mathcal{O}_{CA}^{\top})$$where $$\mathcal{O}^{\top}_{CA}=\begin{bmatrix}C \\ CA \\ \ddots \\ CA^{n-1} \end{bmatrix}$$ is the Observability Matrix. Finally We say the pair $(C,A)$ is Observable if and only if $$\text{rank}(\mathcal{O}_{CA}^{\top})=n$$

Theorem (Minimial realization of LTIC system)

Suppose that $(A,B,C,D)$ with $A\in M_{n}(\mathbb{R})$ is a Realization of some Weighting Pattern. Then $(A,B,C,D)$ is Minimal if and only if $(A,B)$ is Controllable and $(C,A)$ is Observable.