# Obtaining the analytical expression using the state transformation matrix

I am trying to derive the analytical expression in the box below. T is the state transformation matrix. x_dot=A*x(t) is a second order linear system. A has 2 distinct eigenvalues s1 and s2 and diagonalizable.

k1 and k2 depend on initial state x0

I am not able to understand what linear algebra steps have been taken so that e^(s1*t) and e^(s2*t) get decoupled? It doesn't make sense to me because the middle term is diagonal but the terms T and inverse(T) are not diagonal.

• What is the matrix $A$? If it is diagonalizable, then you can take advantage of this and uncouple the system, as a result, obtaining the solution is straightforward. – CroCo Mar 26 '18 at 6:32
• A has 2 distinct eigenvalues s1 and s2 and diagonalizable. But how does it effect T then ? – aadil095 Mar 26 '18 at 19:38
• T is the model matrix not the transition matrix as you claim. The model matrix is a matrix whose columns are the eigenvectors. – CroCo Mar 26 '18 at 21:07
• sorry about that, its state transformation matrix, I have made the change. yes I am aware of the modal matrix. But how do I find out the eigenvectors when I don't know A? I want to multiply and see it getting decoupled. – aadil095 Mar 26 '18 at 21:36
• You assumed the matrix diagonalizable and its eigenvalues are distinct, then the solution has the form you've provided. If you need a rigorous proof, you may ask the question in MathStack. – CroCo Mar 26 '18 at 21:58