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 nonzero vector v with A v = lambda v. The eigenvalue lambda can stretch, shrink, reverse or send v to zero. I is the identity matrix. Nonzero solutions of (A minus lambda I)v = 0 exist exactly when det(A minus lambda I) = 0. For A = [[a, b], [c, d]], this determinant is (a minus lambda)(d minus lambda) minus bc. A diagonal matrix has its diagonal entries as eigenvalues: [[4, 0], [0, 1]] has 4 and 1.
A = [[6, 2], [0, 4]] has characteristic polynomial lambda squared minus 10 lambda + 24 = (lambda minus 6)(lambda minus 4), so its eigenvalues are 6 and 4. For lambda = 4, A minus 4I = [[2, 2], [0, 0]]. Solve 2x + 2y = 0: v = (1, minus 1) gives A v = (4, minus 4) = 4v. For A = [[2, 1], [0, 3]], v = (1, 0) gives A v = (2, 0) = 2v. Diagonal matrices have their diagonal entries as eigenvalues.
In the plane, reflection in a line through the origin fixes that line (lambda = 1) and reverses its perpendicular direction (lambda = -1). A nonzero shear fixes one line and slides parallel to it, with lambda = 1. Identity rotations and half turns preserve every line through the origin, with eigenvalues 1 and -1 respectively. Other planar rotations about the origin preserve no real line. A quarter turn has characteristic equation lambda squared + 1 = 0, with no real roots.
An eigenspace is the full solution space of (A minus lambda I)v = 0, including zero. Eigenvectors are its nonzero vectors. Nonzero multiples of an eigenvector have the same eigenvalue; multiplying by zero does not give an eigenvector. One vector describes a one-dimensional eigenspace. A higher-dimensional eigenspace needs a basis. For A = 2I, the eigenspace for 2 is the whole plane. Eigenvalues stay the same under a basis change; eigenvector coordinates change.
Solve the characteristic equation for the scale factors, then solve one homogeneous system per factor for the directions.
Lesson and practice questions originally authored for this dataset.
Reading summaries of the Khan Academy article Visualizing linear transformations. Videos by Khan Academy. The line-to-point clarification and practice questions are original. The rewritten readings on eigenvalues and diagonalizing a matrix were written for this dataset.
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