Eigenvalues and Eigenvectors
Sweep a vector around the circle and watch it swing away from its own direction — except at two angles, where it does not turn at all.
The Matrix
Which Directions Survive?
turned 0°White is the probe, orange is where the matrix sends it. Green lines are the eigenvector directions.
Eigenvalues
Cross-checks
Eigenvalues and Eigenvectors
An eigenvector keeps its direction under a linear map; its eigenvalue is the scale factor along that direction.
Finding and interpreting them
Eigenvalues satisfy det(A − λI) = 0; for each eigenvalue, solve (A − λI)v = 0 to find its eigenvectors. Some real matrices have complex eigenvalues, so not every transformation has a real eigenvector direction. Symmetric real matrices are especially useful: they have real eigenvalues and an orthonormal eigenbasis.
PCA uses eigenvectors of a covariance matrix to identify directions of variance. In practice, singular value decomposition is often preferred for numerical stability, especially for rectangular data; see SVD and covariance. Eigenvalues describe scaling along special directions, not every feature of a transformation.
Solve a simple example and interpret multiplicity
For A=[[2,0],[0,3]], solving Av=λv gives eigenvalues 2 and 3. For λ=2, (A−2I)v=0 forces the second component to zero, so every nonzero vector along the x-axis is an eigenvector. For λ=3, the eigenvectors lie on the y-axis. A zero vector is never called an eigenvector because it satisfies Av=λv for every λ and carries no direction.
An eigenvalue tells how the transformation acts on its matching eigenvector: λ=2 doubles it, λ=0 collapses it, and a negative λ reverses its direction as well as scaling it. Eigenvectors are not generally perpendicular, except that a real symmetric matrix admits an orthonormal eigenbasis. A matrix may have repeated eigenvalues and too few independent eigenvectors to diagonalise; complex eigenvalues also occur for real matrices such as a planar rotation. Use eigenvectors to understand invariant directions, but do not assume every vector is one. For arbitrary rectangular matrices, SVD provides the more general directional decomposition.
Try this above
Change the matrix and watch which vectors keep their direction. Compare their stretch factors with the eigenvalue readouts.
For a symmetric matrix, eigenvectors belonging to distinct eigenvalues are perpendicular. This gives a useful orthogonal coordinate system in which the transformation acts by independent scaling along each axis. If A=QΛQᵀ, then repeated application raises eigenvalues to powers: Aᵏ=QΛᵏQᵀ. Eigenvalues with magnitude below one shrink under repetition, while those above one grow. Negative eigenvalues also alternate direction. This interpretation helps analyse dynamics, but numerical eigenvectors can be sensitive when eigenvalues are close.
For diagonalisation A=QΛQ⁻¹, the columns of Q are independent eigenvectors and Λ contains their eigenvalues. This representation can simplify repeated application and matrix powers. It exists only when a full eigenbasis is available; repeated eigenvalues do not guarantee enough independent eigenvectors. For symmetric A, Q can be chosen orthogonal, making Q⁻¹=Qᵀ. Do not use a diagonalisation formula when the matrix is defective or when numerical eigenvectors are poorly conditioned.
Sources and next steps
- MIT OpenCourseWare: Linear Algebra covers eigenvalues, eigenvectors, and symmetric matrices.
- Continue with SVD for a decomposition that also handles rectangular matrices.
Overview
The defining relationship
For a square matrix A, a nonzero vector v is an eigenvector when Av = λv. The scalar λ is its eigenvalue. A map may stretch, shrink, or reverse v, but it does not turn it into a different direction.
For A = [[2, 0], [0, 3]], the coordinate axes are eigenvectors: (1, 0) has eigenvalue 2 and (0, 1) has eigenvalue 3. A vector not on either axis generally changes direction. Eigenvectors are not unique in length; every nonzero multiple of one is also an eigenvector.