CDS - Theory Behind Eigenvalues and Eigenvectors

Remember:

Given a linear system: $\dot{x}=Ax$, we seek solutions of the kind:$x(t)=\vec{v}{e}^{\lambda t}$Where: $\vec{v}$ is the eigenvector, while $\lambda$ is the eigenvalue. We can check:$\dot{x}=\lambda e^{\lambda t}\vec{v}\text{and}\dot{x}=A\vec{v}{e}^{\lambda t}$Since eigenvalues and eigenvectors have this property: $\lambda \vec{v}=A\vec{v}$Suppose we have 2 non-zero eigenvalues $({\lambda }_1,{\lambda }_2)$, with eigenvectors ${\vec{v}}_1$ and ${\vec{v}}_2$, then the general solution will be:$x(t)=\alpha {\vec{v}}_1{e}^{{\lambda }_{1}t}+\beta {\vec{v}}_2{e}^{{\lambda }_{2}t}$Where $\alpha ,\beta \in \mathbb{R}$, and we can check that this is the general solution since we can derivate in time to obtain:$\dot{x}={\lambda }_1\alpha {\vec{v}}_1{e}^{{\lambda }_{1}t}+{\lambda }_2\beta {\vec{v}}_2{e}^{{\lambda }_{2}t}$And since, like we said before: $\lambda \vec{v}=A\vec{v}$, we will have that:$\begin{array}{lc}\dot{x}& ={\lambda }_1\alpha {\vec{v}}_1{e}^{{\lambda }_{1}t}+{\lambda }_2\beta {\vec{v}}_2{e}^{{\lambda }_{2}t}\\ & =\alpha (A{\vec{v}}_1)e^{{\lambda }_{1}t}+{\lambda }_2\beta (A{\vec{v}}_2)e^{{\lambda }_{2}t}\\ & =Ax\end{array}$

---

• For each system that does not include a constant, so for $\dot{x}=Ax+c$ where $c=0$, then $x=0$ will always be a ss.

• Example of an explicit analitic solution for a 2D general case,NOT-IMPORTANT

• Basically we have found a general solution for 2D linear systems in the form $\dot{x}=Ax$, but this is just an example, we will solve systems numerically and not analitically.