eigenvalueseigenvectors

An eigenvalue-eigenvector pair of a square matrix A is a scalar λ and a nonzero vector v satisfying Av = λv. Equivalently, λ is an eigenvalue of A if there exists a nontrivial solution to (A − λI)v = 0. The eigenvalues are roots of the characteristic polynomial det(A − λI) = 0, and the corresponding eigenvectors lie in the null space of A − λI.

Eigenvectors corresponding to distinct eigenvalues are linearly independent, and the eigenvectors for a given eigenvalue span its eigenspace. The algebraic multiplicity of λ is its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace.

A matrix A is diagonalizable over a field if it has a full set of linearly independent eigenvectors; equivalently, there exists an invertible P with P−1AP = D diagonal, where the diagonal entries are the eigenvalues. Then A^k = PD^kP−1. For real symmetric matrices, all eigenvalues are real and eigenvectors associated with distinct eigenvalues are orthogonal (the spectral theorem).

To compute them, solve det(A − λI) = 0 for the eigenvalues, then solve (A − λI)v = 0 for corresponding eigenvectors. Numerically, methods include the QR algorithm and power iteration for dominant eigenpairs. Applications include changing basis, principal component analysis, solving linear differential equations, and analyzing Markov chains, where the stationary distribution is an eigenvector with eigenvalue 1.

Example: for A = [[2,1],[1,2]], the eigenvalues are 3 and 1 with eigenvectors proportional to (1,1) and (1,−1).