Unit 8 of 11

Eigenvalues and Diagonalization

  • 6 lessons
  • Videos 38 to 43

An eigenvector of a matrix is a nonzero vector that the matrix sends to a multiple of itself. This unit finds eigenvalues and eigenspaces and uses them to diagonalize matrices.

The characteristic polynomial gives the eigenvalues, and elimination gives a basis for each eigenspace. A repeated eigenvalue can have fewer independent eigenvectors than its multiplicity as a root, which is the difference between geometric and algebraic multiplicity. A matrix can be diagonalized over the chosen scalars exactly when its eigenspace dimensions add up to its size, and then its powers reduce to powers of its eigenvalues. The unit closes with complex numbers and the complex eigenvalues of a quarter-turn rotation, which has no real eigenvector.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Eigenvalues and Eigenspaces

  1. Eigenvectors and Eigenvalues

    What eigenvectors, eigenvalues and eigenpairs are, how an eigenvector marks an invariant line, what negative and zero eigenvalues mean, and why eigenvectors for distinct eigenvalues are independent.

    Video 38

  2. Calculating Eigenvalues and Eigenspaces

    How the characteristic polynomial and the characteristic equation give the eigenvalues of a square matrix, how elimination gives a basis of each eigenspace, and how trace and determinant check the eigenvalues.

    Video 39

  3. Algebraic and Geometric Multiplicity

    How a repeated eigenvalue can come with a full or a deficient eigenspace, how algebraic and geometric multiplicity differ, why the geometric one is bounded by the algebraic one, and what makes a matrix defective.

    Video 40

Diagonalization

  1. Diagonalization and Matrix Powers

    How an eigenbasis gives a diagonalization, why powers of a diagonalizable matrix reduce to powers of its eigenvalues, how this describes a repeated linear update, and how to test whether a matrix is diagonalizable.

    Video 41

Complex Eigenvalues

  1. Complex Numbers for Linear Algebra

    How complex numbers are built from the imaginary unit, how to add, multiply and divide them, and how the complex plane, the conjugate, the modulus and multiplication by i as a quarter-turn fit together.

    Video 42

  2. Complex Eigenvalues and Real Rotations

    How a quarter-turn gets complex eigenvalues and eigenvectors, why the nonreal eigenvalues of a real matrix come in conjugate pairs, and what diagonalization over the complex numbers and the conjugate transpose mean.

    Video 43