Editing Observability (section)

==== Unobservable subspace ====
The ''unobservable subspace'' <math>N</math> of the linear system is the kernel of the linear map <math>G</math> given by<ref name=":1">Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998</ref><blockquote><math> \begin{align}
G \colon \mathbb{R}^{n} &\rightarrow  \mathcal{C}(\mathbb{R};\mathbb{R}^n) \\
         x(0)  &\mapsto  C e^{A t} x(0)
\end{align} </math></blockquote>where <math>\mathcal{C}(\mathbb{R};\mathbb{R}^n)</math> is the set of continuous functions from <math>\mathbb{R}</math> to <math>\mathbb{R}^n </math>.  <math>N</math> can also be written as <ref name=":1" />

:<math> N = \bigcap_{k=0}^{n-1} \ker(CA^k)= \ker{\mathcal{O}} </math>

Since the system is observable if and only if  <math>\operatorname{rank}(\mathcal{O}) = n</math>,  the system is observable if and only if <math>N</math> is the zero subspace.

The following properties for the unobservable subspace are valid:<ref name=":1" />
*<math> N \subset Ke(C) </math>
*<math> A(N) \subset N  </math>
*<math> N= \bigcup \{ S \subset R^n \mid S \subset Ke(C), A(S) \subset N  \} </math>