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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.
مثال کار شده · eigenvalues of [[2,1],[1,2]]
Eigenvalues of [[2,1],[1,2]]
قدم به قدم
- \det(A - \lambda I) = 0
Eigenvalues are the roots of the characteristic polynomial.
- \det\left[\begin{matrix}2 - \lambda & 1\\1 & 2 - \lambda\end{matrix}\right] = 0
Subtract λ from the diagonal.
- \lambda^{2} - 4 \lambda + 3 = 0
Expand the determinant.
- \left(\lambda - 3\right) \left(\lambda - 1\right) = 0
Factor.
- \lambda = 3, \lambda = 1
Eigenvalues (with multiplicity).
- \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.
- \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.
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چطور: Eigenvalues and eigenvectors
- Eigenvalues are the roots of the characteristic polynomial.
- Subtract λ from the diagonal.
- Expand the determinant.
- Factor.
- Eigenvalues (with multiplicity).
- Solve (A − 1I)v = 0 for a basis eigenvector.
- Solve (A − 3I)v = 0 for a basis eigenvector.
سوالاتي که مردم ميپرسن
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.
بیشتر در Linear Algebra
DeterminantsMatrix inverseMatrix multiplicationRow reductionVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis