Eigenvectors and eigenvalues

An eigenvector of a square matrix A is a nonzero vector v that A only rescales: Av = λv. The scalar λ is its eigenvalue. Any nonzero multiple sv is also an eigenvector, with the same eigenvalue, so we usually look for unit eigenvectors.

Example: for A = diag(4, 7), the vector v = [0, 1]ᵀ gives Av = [0, 7]ᵀ = 7v, so v is an eigenvector with eigenvalue 7.

If A has n linearly independent eigenvectors, stacked as the columns of V, with their eigenvalues in the vector λ, then A = V diag(λ) V⁻¹: the eigendecomposition of A. Not every matrix has one, and some need complex numbers. The eigenvalues tell you useful facts: a matrix is singular if and only if one of its eigenvalues is zero.