What exactly is a defective nodal source

Question

I attain repeated root eigenvalue of $-1$ with eigenvector $\begin{pmatrix}2\\1\end{pmatrix}$

However, I'm uncertain if the general solution would be $c_1\cdot e^{-t}\begin{pmatrix}2\\1\end{pmatrix}$ alone, or would it be $c_1\cdot e^{-t}\begin{pmatrix}2\\1\end{pmatrix} + c_2\cdot t\cdot e^{-t}\begin{pmatrix}2\\1\end{pmatrix}$.

Likewise, when I plot the phase portriat in software, I receive an alert claiming that the phase protriat is a defective nodal source. What is that exactly?

Answer 1

In the case of repeated real roots you need to calculate a different second eigenvector which satisfies the equation

$$\begin{pmatrix}a-\lambda & b\\c & d-\lambda\end{pmatrix}\begin{pmatrix}\rho_1\\\rho_2\end{pmatrix}~=~\begin{pmatrix}\eta_1\\\eta_2\end{pmatrix}$$

where $\begin{pmatrix}\eta_1\\\eta_2\end{pmatrix}$ is the first eigenvector and $\lambda$ is the eigenvalue of the matrix. From there on you can conclude to the general solution of the form

$$\vec{x}~=~c_1\cdot e^{\lambda t}\vec{\eta}+c_2(t\cdot e^{\lambda t}\vec{\eta}+e^{\lambda t}\vec{\rho})$$

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The eigenvalues of the given systems are the repeated real roots $\lambda_{1,2}=-1$. From there one you get the first eigenvector $\begin{pmatrix}2\\1\end{pmatrix}$ as you already did. Now just calculate the second eigenvector and you are done.