A certain system with state \(\mathbf{x}\) is described by the state matrices

\[
\begin{aligned}
& \mathbf{A}=\left[\begin{array}{ll}
-2 & 1 \\
-2 & 0
\end{array}ight], \mathbf{B}=\left[\begin{array}{l}
1 \\
3
\end{array}ight] \\
& \mathbf{C}=\left[\begin{array}{ll}
1 & 0
\end{array}ight], \mathbf{D}=[0] .
\end{aligned}
\]

Find the transformation \(\mathbf{T}\) so that if \(\mathbf{x}=\mathbf{T z}\), the state matrices describing the dynamics of \(z\) are in controllable canonical form. Compute the new matrices \(\mathbf{A}_{z}, \mathbf{B}_{z}, \mathbf{C}_{z}\), and \(\mathbf{D}_{z}\).