Spektralform

Spektralform, or spectral form, is a representation of a linear operator through its spectrum. In linear algebra, it often refers to the diagonalization or spectral decomposition of a matrix or operator.

For a linear operator A on a finite-dimensional complex inner product space, if A is diagonalizable, there exists an invertible S such that A = S Λ S^{-1}, where Λ is diagonal with eigenvalues λ_i along the diagonal. If A is normal, S can be chosen unitary, and A = U Λ U*, with U unitary. The columns of U form an orthonormal eigenbasis.

The spectral theorem provides a canonical form for broad classes of operators. For self-adjoint (Hermitian) operators, all eigenvalues are real and there exists a unitary U with A = U Λ U*, where Λ is real diagonal. For normal operators, eigenvalues may be complex and there exists a unitary U with A = U Λ U*. In finite dimensions, the spectral decomposition can also be written as A = Σ λ_i P_i, where P_i is the projection onto the eigenspace corresponding to λ_i; the P_i are mutually orthogonal and sum to the identity.

For more general (including infinite-dimensional) settings, the spectral theorem expresses A as a spectral integral A = ∫ λ dE(λ), with a projection-valued measure E. This continuous extension leads to the spectral form used in functional calculus.

Applications span solving linear systems and differential equations, quantum mechanics, signal processing, and graph theory, where the spectrum informs stability and dynamics. The term Spektralform is closely related to spectral decomposition and diagonalization, though its precise meaning can vary by mathematical context.