solved differential system of equation

I have a system of N differential equation. For solve the system I tried to diagonalize the matrix, for example consider the next system:

$$\begin{pmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{pmatrix} = \begin{pmatrix} 5 & 4 & 3\\ 2 & 6 &8\\ 1 & 11 & 15\end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix}$$

For solve the system i write the central matrix of the form $$P.DD.P^{-1},$$where P is a eigenvector matrix. To obtain the diagonal matrix in mathematics we use:

ss = {{4, 2, 3}, {2, 6, 8}, {1, 11, 15}};
P = Transpose[Eigenvectors[ss]];
DD=Inverse[P].ss.P


The main problem is that there is not correspondence between the elements of matrix DD and the vector $(\dot{x},\dot{y},\dot{z})$ (system equation), since the elements of DD are arranged in descending order.

How can I relate the elements of the diagonal of the matrix DD, with the variable $(x, y, z)$ and $(\dot{x},\dot{y},\dot{z})$ to which they are linked?.

Thanks

• You have a typo. ss = {{4, 2, 3}, {2, 6, 8}, {1, 11, 15}}; does not match the Latex above it. I used the Latex version. – Nasser Jan 9 '18 at 18:13
• You also can't use D as variable. – Nasser Jan 9 '18 at 18:27
• Unless I'm misunderstanding your question, the "correspondence" you're looking for is simply the matrix P: P gives the similarity transformation that takes you between the original basis $(x,y,z)$ and the basis in which DD is diagonalized. – Michael Seifert Jan 9 '18 at 19:40