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.