Consider a linear ODE system of the following form: $$ \frac {dx} {dt} = Ax $$
In case $A$ has real eigenvectors, I can interpret them as the directions in which the system will move, if the initial value is already a point on the eigenvector. The eigenvalues corresponding to each eigenvector would tell me whether the system approaches the fixed point (negative eigenvalue) or moves away from it (pos. eigenvalue). The plot below shows an example vectorfield for the case
$ A = \left( \begin{array}{cc} 2 & 3\\ 1 & 4 \end{array} \right) $, in which the Jacobian $A$ has eigenvalues $1$ and $5$ and eigenvectors $\left( \begin{array}{c} -3 \\ 1 \end{array} \right)$ (blue) and $\left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ (green).
Plot of real eigenvectors:
My question is how to interpret the eigenvectors in case they are complex? If the Jacobian has complex eigenvectors, this means the eigenvalues are also complex, so the system will oscillate. Do the complex eigenvectors tell me something about the rotational axis of these oscillations?
I have plotted the vectorfield of several linear oscillatory systems and to me it seems like the real part of eigenvector points in the direction of the ellipse in which the system oscillates. Here is the vectorfield plot for the system
$$A = \left( \begin{array}{cc} 0 & -3\\ 1 & -2 \end{array} \right)$$
( eigenvalues are $-1 \pm \sqrt{-2}$). The real part of the eigenvector is shown in blue.
Plot of complex eigenvectors:
If this is correct, how can I interpret the imaginary parts of the eigenvectors? Could you maybe point me out an article or a book in which this is explained?
Best Answer
Intuitively, when the system has real eigenvalues and real eigenvectors, the eigenvectors are the directions in which the flow $\vec{x}$ does not change direction.
When the eigenvalues have imaginary part, hence the eigenvectors contain imaginary part, there are no directions in which the flow $\vec{x}$ does not change direction.
If an eigenvector is $\vec{v}_x+i\vec{v}_y$, with corresponding eigenvalue $L_x+iL_y$, then $$A(\vec{v}_x+i\vec{v}_y)=(L_x+iL_y)(\vec{v}_x+i\vec{v}_y)\\ =(L_x\vec{v}_x-L_y\vec{v}_y)+i(L_x\vec{v}_y+L_y\vec{v}_x)$$
Separate the real and imaginary parts: $$A\vec{v}_x=L_x\vec{v}_x-L_y\vec{v}_y\\ A\vec{v}_y=L_x\vec{v}_y+L_y\vec{v}_x$$
So on the real plane, you can see $\vec{v}_x, \vec{v}_y$ are two axis that the flow rotate around. Intuitively, they are major and minor axis of the ellipse or circle when you have a center as fixed point, or spiral. This is not very rigorous argument, but may give you some insight.