Eigenarvolle

Eigenarvolle is not a standard term in English-language mathematics. In many sources it appears as a transliteration, translation, or misspelling of the well-known concept of eigenvalues, and may reflect linguistic variation rather than a distinct mathematical notion. When encountered, it is usually intended to refer to eigenvalues of a square matrix or linear operator.

Definition: For a square matrix A, a scalar λ is called an eigenvalue if there exists a nonzero vector v such that Av = λv. Equivalently, (A − λI)v = 0, so λ satisfies the characteristic equation det(A − λI) = 0. The corresponding nonzero v is an eigenvector.

Properties: Eigenvalues may be real or complex numbers. 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 nullspace of A − λI. Eigenvalues are invariant under similarity transformations, and a matrix is diagonalizable if it has a full set of linearly independent eigenvectors.

Computation and applications: Eigenvalues are found by solving the characteristic polynomial, or by numerical methods such as QR algorithms and power iteration. They play central roles in diagonalization, matrix exponentials, solving linear differential equations, principal component analysis, vibrational and stability analysis, and quantum mechanics.

See also: Eigenvector, eigenvalue, characteristic polynomial, diagonalization, spectral decomposition.