# Kallman Decomposition

## Some background
- $X_r = R(K_c)$ : Reachable space is the range of the
`controllable matrix`
- $X_{no} = Ker(K_o)$ : Not observable Space is the kernel 
of the `observability matrix`
- $X = X_r \bigoplus X_{nr}$ : Each possible state is sum of 
bases of `reachable` and `not-reachable states`

## Full Decomposition
> [!TIP]
> Follow [Example 12](./Examples/EXAMPLE-12.md/#kallman-full-decomposition) To understand this part
- $X_1 = X_r \cap X_{nr}$ : `Reachable` but `Not-Observable` 
  space
- $X_2 =$ Complement of $X_1$ to cover $X_r$ : Both `Reachable`
  and `Observable`
- $X_3 =$ Complement of $X_1$ to cover $X_{no}$ : Both 
  `Not-Reachable` and `Not-Observable`
- $X_4 =$ Complement of all the others to cover $X$ : Bot 
  `Not-Reachable` and `Observable`

From here we have these blocks:
$$
\begin{align*}
    \hat{A} &= \begin{bmatrix}
        \hat{A}_{11} & \hat{A}_{12} & \hat{A}_{13} & \hat{A}_{14} \\
        0 & \hat{A}_{22} & 0 & \hat{A}_{24} \\
        0 & 0 & \hat{A}_{33} & \hat{A}_{34} \\
        0  & 0 & 0 & \hat{A}_{44}
    \end{bmatrix}\\

    \hat{B} &= \begin{bmatrix}
        \hat{B}_{1} \\ \hat{B}_{2} \\ 0 \\ 0
    \end{bmatrix}\\

    \hat{C} &= \begin{bmatrix}
        0 & \hat{C}_{2} & 0 & \hat{C}_{4}
    \end{bmatrix}
\end{align*}
$$

Now, the eigenvalues of $\hat{A} = \cup_i^4 \hat{A}_{ii}$ and:
- $eig(\hat{A}_{11})$: `Reachable` and `Not-Observable`
- $eig(\hat{A}_{22})$: `Reachable` and `Not-Observable`
- $eig(\hat{A}_{33})$: `Not-Reachable` and `Not-Observable`
- $eig(\hat{A}_{44})$: `Not-Reachable` and `Observable`