I am trying to derive the analytical expression in the box below. $T$ is the state transformation matrix. $\dot{x}=Ax(t)$ is a second order linear system. $A$ has two distinct eigenvalues $s_1$ and $s_2$ and is diagonalizable. The size of $A$ is $2 × 2$, and the size of $x$ and $\hat{x}$ is $2×1$.
Here $k_1$ and $k_2$ depend on the initial state $x(0)$.
I am not able to understand what linear algebra steps have been taken so that $e^{s_1 t}$ and $e^{s_2 t}$ get decoupled? It does not make sense to me because the middle term is diagonal but the terms $T$ and $T^{-1}$ are not diagonal.
Any hints can also do!