Eigenvalues and Eigenvectors
Some directions survive a map unchanged except for scaling. Finding them means solving the characteristic equation, and they explain what the map does far better than its entries do.
What a learner can do afterwards
- Find eigenvalues from the characteristic polynomial and eigenvectors from the null space
- Interpret an eigenvector geometrically for a rotation, reflection or shear
- Give a real matrix with no real eigenvalues and explain the geometry
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An eigenvector is a direction the map only stretches: A v = lambda v for some scalar lambda, the eigenvalue. They belong to the map, so they survive a change of basis and show what it does. To find them, solve det(A minus lambda I) = 0 for lambda, then solve (A minus lambda I)v = 0 for the vectors. Diagonal matrices show the answer directly: [[4, 0], [0, 1]] has eigenvalues 4 and 1.
Work a 2 by 2 fully. A = [[6, 2], [0, 4]] has characteristic polynomial lambda squared minus 10 lambda + 24 = (lambda minus 6)(lambda minus 4), so the larger eigenvalue is 6. And v = (1, 0) with A = [[2, 1], [0, 3]] gives A v = (2, 0) = 2v, an eigenvector with eigenvalue 2.
Geometry explains the zoo. Reflections fix their mirror line and flip the perpendicular one. Shears fix one line and slide parallel to it, with eigenvalue 1 on the fixed direction. Rotations other than half turns fix no real line: the quarter turn has lambda squared + 1 = 0, no real eigenvalues.
Two habits keep you safe. Scaling an eigenvector keeps it an eigenvector, so report one clean vector per eigenspace. And the zero vector never counts, since it stretches trivially under everything. Each solution space is the eigenspace for its eigenvalue.
Solve the characteristic equation for the stretch factors, then solve one homogeneous system per factor for the directions.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.