Solving system of differential equations using matrix exponential

[ChiralSuperfields]

Homework Statement

Please see below

Relevant Equations

Please see below

For this problem,

The solution is,

However, can someone please explain to me where they got the orange coefficient matrix from?It seems different to the original system of the form $\vec x' = A\vec x$ which is confusing me.

Thanks!

 

[DaveE]

This is all about transforming the original matrix A into it's Jordan normal form as the easy way to solve $e^{At}$ (the state transition matrix). But then you have to do the inverse transformation to get it back to the original basis. The reason you find the eigenvectors is to create the Jordan form, which, for simple systems is just a matrix with the e-values on the diagonal.

So, in your case the transform to the Jordan normal form uses the e-vector matrix $H= \begin{bmatrix} 1 & 2\\ 1 & 1 \end{bmatrix}$
What is it's inverse $H^{-1}$ and how would you use it?

https://math24.net/method-matrix-exponential.html
https://sites.millersville.edu/bikenaga/linear-algebra/matrix-exponential/matrix-exponential.html

plus soooo many other versions of this problem on the web.

 

[DaveE]

This is also an excellent video of $e^{At}$.

 

[ChiralSuperfields]

Thank you for your replies @DaveE!

Do you please know why $Φ'(t) = e^{At}Φ(0)$ where $Φ$ is fundamental matrix?

Thanks!