Learning Exercise
The directions a matrix leaves alone: from eigenvectors to phase portraits
Students first find eigenvectors by eye, turning a vector until the matrix sends it back onto its own line, and then explain what they found with det(A − λI). In Part 2 the same eigenvalues decide the shape of the solutions of x′ = Ax. In presentation mode the instructor can use H to hide the eigenvalues and eigenlines while students predict. Check the current hide-answers state, which can persist between pages.
For linear algebra or a first course in differential equations (college lower division).
- Material:
- Bun & Bolt Math: A Visual Dictionary of Mathematics
- Date last modified:
- October 29, 2026
- Created by:
- Ziyue Gan
- License:
-
Bun & Bolt Math is a free visual dictionary of mathematics with more than 100 interactive entries covering algebra,... see more
Exercise
Materials: Bun & Bolt Math, Eigenvector, https://bunandbolt.com/math/eigenvector, and Phase portrait, https://bunandbolt.com/math/phase-portrait
Part 1, Eigenvector page (the default matrix is A = (3 1; 2 2)):
1. Turn the direction of v around the circle and watch Av. Find every direction where Av lies on v's line. Write down each direction and the factor λ. Check by hand: A(1, 1) = 4(1, 1) and A(1, −2) = (1, −2), so λ = 4 and λ = 1.
2. Open the A − λI window and drag λ along the graph of det(A − λI). Ask: what happens to the plane when λ reaches a root? At λ = 4, A − 4I flattens the whole plane onto one line and wipes out the direction (1, 1). Explain why that forces det(A − 4I) = 0, and why solving det(A − λI) = 0 therefore finds the eigenvalues.
3. Press Page Down to reach the highlights "A shear keeps one line" and "A turn keeps none". Before revealing, predict how many directions stay on their own line in each case. Relate these specific examples to a double real root and a pair of nonreal conjugate roots. A double root alone does not imply just one eigenline: compare with the identity matrix, where every line is an eigenline. The rotation shown is neither a zero-degree nor a 180-degree turn.
4. Back on the default matrix, start v away from both eigen lines and press Apply repeatedly. Predict where the direction of v, Av, A²v, … ends up, and explain why with λ = 4 and λ = 1.
Part 2, Phase portrait page:
5. Drag the point on the trace–determinant map into each region and sketch the picture you see. Write τ = tr(A), Δ = det(A), and D = τ² − 4Δ. Fill in a table with columns for Δ, τ, D, eigenvalue type, and portrait type. Use Δ < 0 for a saddle; Δ > 0 and D > 0 for a node; Δ > 0, D < 0 and τ ≠ 0 for a spiral; and Δ > 0, τ = 0 for a center. For nodes and spirals, τ < 0 means inward motion and τ > 0 means outward motion. Leave Δ = 0 and D = 0 as boundary cases for discussion: a zero eigenvalue gives equilibrium points, and repeated eigenvalues require inspecting whether A is a scalar matrix.
6. Use the highlight "Saddle". Identify the two invariant lines, each of which contains a family of solutions. Explain why a solution along an eigenvector v has the form x(t) = c e^{λt} v, and why the solution moves toward the origin along one line and away along the other.
7. Use the highlight "Out slow, then fast". For a solution off both eigenlines, predict its direction as t tends to −∞ (approaching the origin) and as t tends to +∞. Here λ = 1 and 2: the slow line y = x dominates near the origin and the fast line y = 2x dominates far away. The zero solution itself never leaves the origin.
Audience
Technical Notes
Runs in a current web browser on computers, tablets and phones. No account or installation. On touch screens, press Interact on the graph to work it with a finger.
Requirements
Multiplying a 2 × 2 matrix by a vector, the determinant of a 2 × 2 matrix, solving a quadratic equation. For Part 2, students should also know derivatives and the solution of x′ = λx; otherwise provide a short introduction before starting Part 2.
Topics
Eigenvectors and eigenvalues; determinants; linear systems of differential equations; trace and determinant; nodes, saddles, spirals and centers.
Type of Task
Learning Objectives
1. Recognize an eigenvector as a direction that a matrix sends along its own line, and read the eigenvalue as the factor that stretches, shrinks or reverses it.
2. Explain why the eigenvalues are the roots of det(A − λI).
3. Compare the examples with two real eigenlines, one real eigenline (a nontrivial shear), and no real eigenline (the rotation shown). Recognize that a scalar multiple of the identity has every line as an eigenline.
4. Predict the kind of phase portrait of x′ = Ax using trace τ, determinant Δ and discriminant τ² − 4Δ, and identify its invariant lines with real eigenvectors. At repeated eigenvalues, distinguish a scalar matrix from a defective one.
Assessment
Questions to hand in:
1. In your own words, what is an eigenvector, and what does its eigenvalue tell you?
2. Why are the eigenvalues of A exactly the values of λ that make det(A − λI) = 0?
3. Give the trace and determinant of a matrix whose phase portrait is a spiral moving inward, and explain how you chose them.