What is the characteristic equation for the eigenvalues $\lambda$?

When you use the quadratic formula to solve for $\lambda$, what number do you get inside the square root?

Since this number is

, when you take the square root, you get an imaginary number with a factor of $i=\sqrt{-1}$. The result is that you get complex numbers for the eigenvalues. The eigenvalues should look something like $1.7 + 0.3i$ and $1.7 - 0.3i$ (but with different numbers other than $1.7$ and $0.3$).

What are the eigenvalues? $\lambda_1=$

, $\lambda_2=$

The fact that eigenvalues are complex is the reason why the solution is rotating around. Remember, for real distinct eigenvalues, we decomposed the initial condition into a linear combination of the eigenvectors. Each of those eigenvector terms were multiplied by the corresponding eigenvalue at each time step. Although the same thing is true for the case with complex eigenvalues, we aren't going to get into calculating complex eigenvectors. (If you want to see them, you can calculate them with R or another computer program.)

Rather than worrying about complex eigenvectors, we'll instead just focus on the fact that, if the eigenvalues are complex, then they don't have real eigenvectors. For two-dimensional systems, that means there are no directions in the phase plane (such as the axes graphed above) where the solution is just scaled; i.e., there is no direction that, if the solution points in that direction, it will stay there forever. As a result it just rotates around, always changing direction.

The next question is, as the solution is rotating around, is it growing or shrinking, i.e., is it getting further away or closer to the origin? If we had a real eigenvalue $\lambda$, when we multiply by $\lambda$ each time step, we know the solution will grow if $|\lambda| > 1$ and will shrink if $|\lambda| < 1$. If we really wanted to do the analysis for complex eigenvalues, we'd discover that something similar is happening even in that case (i.e, that multiplication by a complex eigenvalue $\lambda$ is still involved at each time step). We're not going to worry about arithmetic with complex numbers, but only state that the same conclusion is true. The solution will be growing if $|\lambda| > 1$ and will be shrinking if $|\lambda| > 1$.

What does the absolute value sign mean for complex numbers? It's the “magnitude” or “modulus” of the complex number. The formula for the modulus is the same as that for the magnitude of a two-dimensional vector. If $\lambda = a+ib$ for two numbers $a$ and $b$, then its modulus, or magnitude, is $|\lambda| = \sqrt{a^2+b^2}$.

You show have found two complex eigenvalues $\lambda_1$ and $\lambda_2$. What are the moduli are of the eigenvalues?

$|\lambda_1|=$

, $|\lambda_2|=$

(The fact that they have the same modulus follows from the fact that they differ only in the sign of their imaginary component, i.e., the eigenvalues are complex conjugates of each other.)

The eigenvalue magnitudes $|\lambda_1|$ and $|\lambda_2|$ are

. Therefore the solution is

even as it is rotating.

##### Hint

In R, you can use the command

`abs` to compute the modulus of a complex number.

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