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Eigenvalues and Eigenvectors

UniversityMathematicsEigenvalues are the roots of λ² - (tr A)λ + det A = 0; eigenvectors solve (A - λI)v = 0

About this experiment

A matrix moves almost every vector to a new direction. A few special directions are only stretched, shrunk or flipped: these are the eigenvectors, and the stretch factors are the eigenvalues. Turn a vector v around and watch its image Av; when the two line up, you have found an eigenvector. Compare symmetric matrices, shears, rotations and projections, and use power iteration to see how repeated multiplication finds the dominant eigenvector by itself.

Learning goals

  • State the eigenvalue equation Av = λv and interpret it geometrically
  • Find the eigenvalues of a 2×2 matrix from its characteristic polynomial
  • Relate the trace and determinant to the sum and product of the eigenvalues
  • Recognise matrices with complex, repeated or defective eigenvalues
  • Explain why power iteration converges to the eigenvector with the largest |λ|

How to use it

  1. 1With the symmetric example, drag v until Av points the same way. Read λ. Find the second eigen-direction. What angle is between them?
  2. 2Press Sweep v around and watch the strip below. How many zero crossings are there in 180°, and where are they?
  3. 3Choose Rotation. Why can no direction stay fixed? Read the complex eigenvalues.
  4. 4Choose Shear. How many eigen-directions are there? Look at the discriminant.
  5. 5Choose Not symmetric. Are the eigenvectors still perpendicular?
  6. 6Point v in any direction and press Power iteration step several times. Which eigenvector does v approach, and how fast?

The model

Definition of an eigenvector v with eigenvalue λ
Characteristic polynomial of a 2×2 matrix
Trace and determinant from the eigenvalues
Δ > 0: two real eigenvalues; Δ = 0: repeated; Δ < 0: complex pair
Power iteration and its rate of convergence

Assumptions

  • All entries are real. Eigenvectors are shown as unit vectors; any non-zero multiple is an eigenvector too.
  • Two eigenvalues closer than 10⁻⁹ in the discriminant are treated as repeated. A repeated eigenvalue has one eigen-direction (defective) unless A is a multiple of the identity.
  • The probe vector v always has length 1, so the length of Av shows how much A stretches that direction.

Try this in class

  1. Q1Find the eigenvalues and eigenvectors of A = [[4, 1], [2, 3]].
  2. Q2Show that a rotation by any angle other than 0° or 180° has no real eigenvectors.
  3. Q3A matrix has trace 5 and determinant 6. What are its eigenvalues?
  4. Q4Why are the eigenvectors of a symmetric matrix always perpendicular?
  5. Q5Power iteration converges slowly when λ₂ / λ₁ is close to 1. Test this with the matrix [[2, 0], [0, 1.9]].

Frequently asked questions

What does a negative eigenvalue mean?

The eigenvector is flipped to point the opposite way, then scaled by |λ|. The reflection example has eigenvalues 1 and -1: vectors on the mirror line stay put, vectors perpendicular to it are flipped.

Where are eigenvectors used?

Everywhere a system is repeated or vibrates: normal modes of molecules and bridges, stability of equilibria in differential equations, principal component analysis in statistics, quantum states, and the original PageRank algorithm, which is power iteration on the web link matrix.

What does the strip below the plane show?

For each direction of v it plots the cross product v × Av, which is zero exactly when Av is parallel to v. Each zero crossing is an eigen-direction. If the curve never crosses zero, there are no real eigenvectors.