Eigenvalues

An eigenvalue of a square matrix A is a scalar λ for which there exists a nonzero vector v satisfying Av = λv, where v is called an eigenvector associated with λ. Equivalently, λ is an eigenvalue if the linear transformation A has a nontrivial invariant direction, i.e., a nonzero vector that is mapped to a scalar multiple of itself.

To find eigenvalues, solve det(A − λI) = 0; the polynomial p_A(λ) = det(A − λI) is the characteristic polynomial, and its roots are the eigenvalues (possibly complex).

Eigenvectors are obtained by solving (A − λI)v = 0 for each eigenvalue λ. The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial; the geometric multiplicity is the dimension of its eigenspace. Generally geometric multiplicity ≤ algebraic; a matrix is diagonalizable if the sum of geometric multiplicities equals the dimension of the matrix.

Two basic identities relate eigenvalues to matrix structure: trace(A) equals the sum of the eigenvalues (counting multiplicities), and det(A) equals the product of the eigenvalues. Real matrices can have complex eigenvalues, which occur in conjugate pairs.

Special cases: real symmetric matrices have real eigenvalues and orthogonal eigenvectors; more generally, Hermitian matrices have a full set of orthonormal eigenvectors (the spectral theorem).

Computation: exact solutions via the characteristic polynomial are feasible for small matrices; for larger or ill-conditioned problems, numerical methods such as the QR algorithm, power iteration, and inverse iteration are used. Eigenvalues can be sensitive to perturbations.

Applications span solving systems of differential equations and stability analysis, vibrations, quantum mechanics, PageRank, and dimensionality reduction methods like principal component analysis, among others.