Linear Algebra · Chapter 09

Eigenvalues and eigenvectors

The directions a map only stretches: eigenvalues from the characteristic polynomial, eigenspaces and the two multiplicities, diagonalisation, and what eigenvalues say about powers and the long run.

A v = λ v

det (A - λ I) = 0

A = P D P^{- 1}

E_{λ} = ker (A - λ I)

v \neq = 0

A^{k} = P D^{k} P^{- 1}

(λ_{1} 0 0 λ_{2})

(A - λ I) v = 0

Lessons

Build the foundation

01Eigenvectors and eigenvaluesMost vectors are turned by a linear map as well as stretched. A few special directions are only stretched, or shrunk, or flipped, and along them the map is as simple as multiplying by a number. Those directions are the eigenvectors, and the numbers are the eigenvalues.Read lesson →
02The characteristic polynomialThe eigenvalues are the numbers that make a certain determinant zero, and that determinant, written out, is a polynomial. Finding the eigenvalues of a matrix is finding the roots of that polynomial, and then everything the course knows about polynomials applies.Read lesson →
03Eigenspaces and multiplicityFor one eigenvalue there is usually not just one eigenvector but a whole subspace of them, a line or a plane that the map stretches by the same factor. How large that subspace is, compared with how often the eigenvalue repeats as a root, decides whether the map can be described simply.Read lesson →
04DiagonalisationIf a map has enough eigenvectors to make a basis, then in that basis it does nothing but stretch each basis vector by its eigenvalue, and its matrix is diagonal. Diagonalising is finding that basis, and it is possible exactly when no eigenvalue is short of eigenvectors.Read lesson →
05Powers and long-run behaviourApplying a map again and again stretches each eigenvector by its eigenvalue again and again. Eigenvalues larger than one in size win, those smaller than one fade away, and the long-run behaviour of a repeated process can be read off the eigenvalues without computing a single power.

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06Chapter practiceBring the chapter together: find eigenvalues and eigenvectors, compare the two multiplicities, decide whether a matrix diagonalises, and use the eigenvalues to predict powers. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Eigenvalues and eigenvectors

Build the foundation