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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.

İşlənmiş nümunə · 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]

Addım-addım

  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.

Cavabı göstər
\lambda = 3,\; \lambda = 1

İndi siz Problem seçin, ya da özünü yazın ya da çəkin. Hər addım, bir rəsm, cavab siz soruşana qədər gizli qalır.

Özün işlə

Pulsuz hesab hər dərsdə qeydlər əlavə edir, bitirdiyiniz işlərin qeydini saxlayır, həll etdiyiniz problemləri bir yerdə saxlayır və bu səhifə haqqında soruşa biləcəyiniz müəllimi də əlavə edir. Riyaziyyat özü daxil olub-olmadığınıza baxmayaraq hər kəsə açıqdır.

Qeyd Et Giriş

Burada istifadə edilən simvollar

Tam təsviri, şəkli və hər hərfin nə demək olduğunu görmək üçün hər hansı simvolu vurun.

Növbəti: 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.

İnsanlar soruşurlar

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.

Daha çox Linear Algebra

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