List of top Control Systems Questions on State Space Analysis

A system is characterized by the following state equation and output equation (\(u\): input, \(\mathbf{x}\): state vector, \(y\): output)
\[ \dot{\mathbf{x}}=\begin{bmatrix}a&b\\-a&0\end{bmatrix}\mathbf{x}+\begin{bmatrix}1\\0\end{bmatrix}u \]
\[ y=\begin{bmatrix}1&2\end{bmatrix}\mathbf{x} \]
What are the values of \(a\) and \(b\) for which the poles of the transfer function are at \(-2+j3\) and \(-2-j3\)?

A system is represented in state-space form as follows (\(u\): input, \(\mathbf{x}\): state vector, \(y\): output)
\[ \dot{\mathbf{x}}=\begin{bmatrix}1&2\\-3&0\end{bmatrix}\mathbf{x}+\begin{bmatrix}1\\2\end{bmatrix}u \]
\[ y=\begin{bmatrix}1&2\end{bmatrix}\mathbf{x} \]
Consider the new state vector \(\mathbf{z}=\begin{bmatrix}2&1\\-1&0\end{bmatrix}\mathbf{x}\)
What is the state-space representation of the system in terms of the new state vector \(\mathbf{z}\)?