eigenarvot

Eigenarvot, or eigenvalues, are scalars associated with a square matrix that describe how certain vectors are scaled under a linear transformation. If A is a square matrix and x is a nonzero vector such that Ax = λx, then λ is an eigenvalue of A and x is a corresponding eigenvector. The equation Ax = λx can be rearranged to (A − λI)x = 0, which leads to the characteristic equation det(A − λI) = 0. The eigenvectors form the eigenspaces, and the multiplicities of eigenvalues are classified as algebraic (multiplicity as a root of the characteristic polynomial) and geometric (dimension of the corresponding eigenspace).

Key properties include that a matrix is diagonalizable if it has n linearly independent eigenvectors. Real symmetric (or Hermitian) matrices have real eigenvalues and can be orthogonally diagonalized, with eigenvectors that are mutually orthogonal. The eigenvalues of A^T are the same as those of A, and eigenvectors corresponding to distinct eigenvalues are linearly independent.

Computation typically starts with solving the characteristic polynomial det(A − λI) = 0 to find eigenvalues, followed by solving (A − λI)x = 0 to obtain eigenvectors. Numerically, methods such as the QR algorithm, power iteration (which finds the dominant eigenvalue), and inverse iteration are used for larger matrices.

Example: For A = [[3, 1], [0, 2]], the eigenvalues are 3 and 2. An eigenvector for λ = 3 is (1, 0), and for λ = 2 is (1, −1). Applications span linear differential equations, computer graphics, and data analysis techniques such as principal component analysis, where eigenvalues indicate the variance explained by principal components.

See also: eigenvectors, characteristic polynomial, diagonalization, spectral theorem.