I have a state space representation, system S1, in the form of:
$ \frac{dx}{dt} = Ax + bu $ $y = c^Tx$
This system is transformed with the state transform $x=T z$ into the system S2: $ \frac{dz}{dt} = \begin{pmatrix} -1 & -2 \\-1 & -2 \end{pmatrix}z + \begin{pmatrix} 0 \\1 \end{pmatrix}u $ $y = \begin{pmatrix} 1 &-1 \end{pmatrix}z $ T is the transformation matrix.
The only thing I know from system S1 is, that is in diagonal form.
So $A$ should look something like this: A = $\begin{pmatrix} a & 0 \\0 & d \end{pmatrix}$
I think I know how I could transform S1 into S2 but I don't know the other way.
Found some formulas like $T^{-1}AT = \begin{pmatrix} -1 & -2 \\-1 & -2 \end{pmatrix}$ and $T^{-1}b = \begin{pmatrix} 0 \\1 \end{pmatrix}$ I have a problem obtaining the transformation matrix.
I also thought about the eigenvalues of S2 (0, -3), but I don't know if I am right that these values are the diagonal points of the matrix A.
edit:
So with @Sasha 's help I got the system matrix A:
$p_1 = \begin{pmatrix} -2 \\ 1 \end{pmatrix}$ $p_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ $T = \begin{pmatrix} -2 & 1 \\1 & 1 \end{pmatrix}$ $T^{-1}\begin{pmatrix} -1 & -2 \\-1 & -2 \end{pmatrix}T = \begin{pmatrix} 0 & 0 \\0 & -3 \end{pmatrix}$ $b = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ $c = \begin{pmatrix} -\frac{2}{3} \\ -\frac{1}{3} \end{pmatrix}$
Some additional infos:
Matrix A
Matrix A is the system matrix, and relates how the current state affects the state change x' . If the state change is not dependent on the current state, A will be the zero matrix. The exponential of the state matrix, eAt is called the state transition matrix.
Matrix B
Matrix B is the control matrix, and determines how the system input affects the state change. If the state change is not dependent on the system input, then B will be the zero matrix.
Matrix C
Matrix C is the output matrix, and determines the relationship between the system state and the system output.