Eig

Eig, short for eigenvalue and eigenvector, refers to a pair consisting of a scalar λ and a nonzero vector v such that Av = λv for a square matrix A. The scalar λ is called an eigenvalue and v the corresponding eigenvector. The collection of eigenvalues is the spectrum of A, and they satisfy the characteristic equation det(A − λI) = 0.

Eigenvalues can be real or complex, and eigenvectors may be real or complex as well. The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the corresponding eigenspace. A matrix is diagonalizable if it has a full set of linearly independent eigenvectors; otherwise, it may require a Jordan form.

For real symmetric (or Hermitian) matrices, eigenvalues are always real and eigenvectors corresponding to distinct eigenvalues are orthogonal, enabling a simple spectral decomposition A = QΛQ^T. In general, A can be decomposed as A = VΛV^{-1} when diagonalizable, or expressed via Jordan form otherwise.

Numerically, eigenvalues are found by algorithms such as the QR algorithm; common software provides routines named eig or linalg.eig. For example, MATLAB returns [V,D] = eig(A), with D diagonal containing eigenvalues and V containing eigenvectors. NumPy's numpy.linalg.eig returns w, v such that Av = wv. For large sparse matrices, iterative methods such as power iteration or the Arnoldi method are used.

Applications include stability analysis of dynamical systems, solving linear differential equations, Markov chains, and principal component analysis, where the eigenvectors indicate principal directions of variance.