Complex Eigenvalues
Differential Equations · Section 7.6 · generated practice set
Complex eigenvalues, real solutions from Euler's formula, and the spiral sink / source / centre split.
Practice this set → Fresh numbers on every attempt. No account needed.
Method
When $det(A - lambda I) = 0$ has complex roots $lambda = alpha pm ieta$, the algebra is the same but the picture rotates.
Building real solutions
Take the eigenvector $mathbf{v} = mathbf{a} + imathbf{b}$ for $lambda = alpha + ieta$ (the conjugate pair gives nothing new). Then $$mathbf{x} = e^{alpha t}Big[c_1left(mathbf{a}coseta t - mathbf{b}sineta t ight) + c_2left(mathbf{a}sineta t + mathbf{b}coseta t ight)Big].$$ In practice: compute $mathbf{v}e^{lambda t}$, expand with Euler's formula, and take the real and imaginary parts as your two real solutions.
The phase portrait
- $alpha < 0$ → spiral sink, asymptotically stable — inward spiral.
- $alpha > 0$ → spiral source, unstable — outward spiral.
- $alpha = 0$ → center — closed orbits. Stable, but not asymptotically stable: trajectories stay near the origin without approaching it.
$eta$ sets how fast the trajectory winds around and never affects stability. The direction of rotation comes from the sign of the off-diagonal entries, most easily found by evaluating $Amathbf{x}$ at a convenient point such as $egin{pmatrix} 1 \ 0end{pmatrix}$ and seeing which way the arrow points.
Definitions and theorems
Worked example
Classify the origin for $mathbf{x}' = egin{pmatrix} -1 & -4 \ 1 & -1end{pmatrix}mathbf{x}$.
- $ ext{tr}A = -2$, $det A = 1 + 4 = 5$, so $lambda^2 + 2lambda + 5 = 0$.
- Discriminant $4 - 20 = -16 < 0$, so $lambda = dfrac{-2 pm 4i}{2} = -1 pm 2i$.
- $alpha = -1 < 0$ and $eta = 2$.
- The origin is a spiral sink: trajectories wind around it while decaying like $e^{-t}$.
- They complete one full turn every $2pi/eta = pi$ units of time, shrinking by a factor $e^{-pi} approx 0.043$ each turn.
Common mistakes
- Using both conjugate eigenvalues. One of the pair gives both real solutions; the other repeats them.
- Letting $eta$ decide stability. Only the sign of $alpha$ matters.
- Calling a center asymptotically stable. Orbits are closed — they stay nearby but never approach.
- Leaving $i$ in the final answer. The general solution of a real system is real; the complex arithmetic is scaffolding.
- Guessing the spiral direction from the eigenvalues. They do not encode it — test the vector field at a point.
Practice it
Reading the method is not the same as being able to run it under time pressure. This set generates a new problem with new coefficients every attempt, marks each part separately, and shows the full worked solution.