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Eigenvalues and eigenvectors

The characteristic polynomial and the directions a matrix only stretches.

An eigenvector is a direction a matrix does not rotate, only stretches by its eigenvalue. Find eigenvalues as roots of det(A − λI), then eigenvectors by solving (A − λI)v = 0. In the 3D view the eigenvectors are drawn on the transformed cube.

Arbejdstxrd eksempel · eigenvalues of [[2,1],[1,2]]

Linear-Algebra · Eigenvalues

Eigenvalues of [[2,1],[1,2]]

\left[\begin{matrix}2 & 1\\1 & 2\end{matrix}\right]

Trin for trin

  1. \det(A - \lambda I) = 0

    Eigenvalues are the roots of the characteristic polynomial.

  2. \det\left[\begin{matrix}2 - \lambda & 1\\1 & 2 - \lambda\end{matrix}\right] = 0

    Subtract λ from the diagonal.

  3. \lambda^{2} - 4 \lambda + 3 = 0

    Expand the determinant.

  4. \left(\lambda - 3\right) \left(\lambda - 1\right) = 0

    Factor.

  5. \lambda = 3, \lambda = 1

    Eigenvalues (with multiplicity).

  6. \lambda = 1:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}-1\\1\end{matrix}\right]

    Solve (A − 1I)v = 0 for a basis eigenvector.

  7. \lambda = 3:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}1\\1\end{matrix}\right]

    Solve (A − 3I)v = 0 for a basis eigenvector.

Afslør svaret
\lambda = 3,\; \lambda = 1

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Hvordan: Eigenvalues and eigenvectors

  1. Eigenvalues are the roots of the characteristic polynomial.
  2. Subtract λ from the diagonal.
  3. Expand the determinant.
  4. Factor.
  5. Eigenvalues (with multiplicity).
  6. Solve (A − 1I)v = 0 for a basis eigenvector.
  7. Solve (A − 3I)v = 0 for a basis eigenvector.

Spørgsmål folk spørger

What does a determinant mean geometrically?

It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.

What is an eigenvector?

A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.

Why is matrix multiplication not commutative?

Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.

Mere i Linear Algebra

DeterminantsMatrix inverseMatrix multiplicationRow reductionVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis