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M03.1 · Eigenvalues and eigenvectors

A free, illustrated module of 4 lessons and one interactive in M03 Linear Algebra II: Decompositions: Vectors that stay on their line; The characteristic polynomial; Trace, determinant and eigenspaces; Complex eigenvalues: a rotation hidden inside.

M03.1 L2 · Intermediate 4 lessons · about 100 min Live · in review

Part of M03 Linear Algebra II: Decompositions in the Mathematics track.

Most vectors are knocked off their line by a matrix. A few are not: the map only stretches, shrinks or flips them, and those few directions explain what the matrix does. These four lessons define eigenvectors by that picture, find them through the characteristic polynomial and a null space, read the trace and determinant off the eigenvalues, separate a repeated eigenvalue with a whole plane of eigenvectors from one with a single line, and meet the complex eigenvalues of a map that turns every vector: a rotation and a stretch hidden in a skewed basis.

What you need first. Matrices as maps of the plane (M02.4), powers of a matrix (M02.5), the null space (M02.7), the determinant and its link to collapse (M02.8), change of basis and similar matrices (M02.9), and, for the last lesson, complex numbers in polar form and quadratics with a negative discriminant (M01.9).

Lessons

1. Vectors that stay on their line M03.1.1 25 min

   Most vectors are knocked off their line by a matrix, but a few are only stretched, shrunk or flipped. Meet eigenvectors and eigenvalues as a picture, check a claimed eigenvector with one multiplication, and find them by sweeping round the circle in code.

2. The characteristic polynomial M03.1.2 25 min

   A number is an eigenvalue exactly when A minus lambda times I flattens space. Turn that into the equation det(A - lambda I) = 0, solve it for the running matrix, find each eigenvector as a null space, and meet numpy.linalg.eig.

3. Trace, determinant and eigenspaces M03.1.3 25 min

   The eigenvalues add up to the trace and multiply to the determinant. See why, collect the eigenvectors of one eigenvalue into a subspace, tell a repeated eigenvalue with a plane of eigenvectors from one with a single line, and prove that different eigenvalues give independent eigenvectors.

4. Complex eigenvalues: a rotation hidden inside M03.1.4 25 min

   A map that turns every vector has no real eigenvector, but its characteristic equation still has roots: a complex conjugate pair. Read their modulus as a stretch and their argument as a turn, find the basis in which the running map T is a plain turn and stretch, and see why repeated application spirals.

The interactive

M03.1-X1 · Eigenvector hunter. Pick a 2 by 2 matrix or set its entries, then turn a unit vector around the circle. Its image is drawn beside it, and when the two line up the line lights up and the read-out gives the stretch factor, or says that no direction ever lines up. It lives in Vectors that stay on their line.