maths.free › Linear Algebra › Eigenvalues and eigenvectors

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.

작업 예제 · 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]

단계별로

  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.

답을 드러내세요
\lambda = 3,\; \lambda = 1

이제 너 문제를 선택하거나, 타이핑하거나, 자신만의 문제를 그려보세요. 모든 단계는 그림으로 표시되며, 답은 질문을 할 때까지 숨겨져 있습니다.

자신의 작업을 유지

무료 계정은 모든 수업에 노트를 추가, 당신이 완료 한 기록, 당신이 해결 한 문제를 한 곳에, 그리고 당신이이 페이지에 대해 물어 수업 지도자. 수학 자체는 모두에게 열려 있습니다, 로그인 여부.

가입하기 로그인

여기서 사용된 기호

기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.

어떻게: 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.

사람들이 묻는 질문

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.

에 더 Linear Algebra

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