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Labs / Mathematics
Eigenvectors
A 2×2 matrix picks up the blue vector v and moves it to a new place, M·v. For most directions the arrow twists to point somewhere new — but along a few special directions the matrix only stretches or flips the vector without turning it at all. Sweep varound and hunt for those directions: they are the matrix's eigenvectors, and the stretch factors are its eigenvalues.
Labs / Mathematics / Eigenvectors
Eigenvectors
A 2×2 matrix picks up the blue vector v and moves it to a new place, M·v. For most directions the arrow twists to point somewhere new — but along a few special directions the matrix only stretches or flips the vector without turning it at all. Sweep varound and hunt for those directions: they are the matrix's eigenvectors, and the stretch factors are its eigenvalues.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
eigenvalues 1.55, 0.65v ∠ 20°stretch 1.00×turn 0°
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- Drag v slowly all the way around. How many directions make the “turn” readout snap to 0° or 180°?
- When the two arrows line up, does the stretch factor match one of the eigenvalues in the readout?
- Turn on “Iterate Mⁿv” from a random start. Which eigenvector does the vector drift toward, and why that one?
- Load the Rotation preset. Why can no real direction ever stay parallel — where did the eigenvectors go?
- Set b and c equal (a symmetric matrix). What's special about the angle between the two eigenvector lines?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.