I'm trying to determine observability of the following system:
$$ \mathbf{A} = \begin{bmatrix}-10 & 0 & -10 & 0 \\0 & -1 & 9 & 0\\0&-1&-1&0\\1&0&0&0\end{bmatrix} $$ $$ \mathbf{B} = \begin{bmatrix}20&3\\0&-3\\0&0\\0&0\end{bmatrix} $$ $$ \mathbf{C} = \begin{bmatrix}1&0&0&0\end{bmatrix}; \mathbf{D} = \begin{bmatrix}0\end{bmatrix} $$
I found the observability matrix to be: $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ -10 & 0 & -10 &0\\ 100 & 10 & 110 & 0\\ -100 &-120&-1020&0 \end{bmatrix} $$ Which has the RREF form of: $$ \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&0 \end{bmatrix} $$
Obviously the system is not completely observable, and I was able to determine through the $\mathbf{A}$ and $\mathbf{C}$ matrices that it would be the fourth state variable that is not observable, but I was unclear as to whether this could be determined from the observability matrix. Is it the fourth column being all zeroes that would tell me this, or is it the fourth row which would tell me this? Would it be the same for a controllability matrix?
In general you can find the null space of the observability matrix, whose basis should also be eigenvectors of $A$. So, the eigenvalues that correspond to those eigenvectors are the unobservable states.