nilaieigen

Nilai eigen, often written as eigenvalue in English, is a scalar λ associated with a square matrix A for which there exists a nonzero vector v such that Av = λv. Equivalently, λ is a root of the characteristic polynomial det(A − λI) = 0, and the corresponding vectors v are called eigenvectors. For a real matrix A, eigenvalues may be real or occur in complex conjugate pairs; over the complex field, every square matrix has eigenvalues.

Analytically, eigenvalues for a 2×2 or small matrices can be found by solving the characteristic polynomial. For larger matrices, numerical methods are used, such as the QR algorithm, power iteration, inverse iteration, Jacobi’s method, and sparse solvers like Lanczos. Diagonalization occurs when there exists a full set of linearly independent eigenvectors; in this case A can be written A = PDP^{-1}, with D diagonal containing eigenvalues.

Properties and applications: eigenvalues can be real or complex, and their multiplicities may differ from the number of independent eigenvectors. If a matrix is diagonalizable, many matrix functions become straightforward to compute via its eigen decomposition. Nilai eigen play a central role in various fields, including stability analysis in differential equations, principal component analysis in statistics, vibration analysis in engineering, and quantum mechanics. Complex eigenvalues correspond to oscillatory modes in dynamic systems.

Example: for A = [[4,1],[2,3]], the eigenvalues are 5 and 2, with corresponding eigenvectors [1,1] and [1,−2], respectively.

Origin: the term eigen is from German, meaning “own” or “characteristic.” The concept and terminology emerged in 19th-century linear algebra, with contributions from multiple mathematicians and widespread use in mathematical literature.