spektriteoreemid

Spektriteoreemid (spectral theorems) are results in linear algebra and functional analysis that describe how operators relate to their spectra and can be represented via spectral decompositions. They are central to understanding linear operators on inner-product spaces, especially normal and self-adjoint ones.

In finite dimensions over the complex field, the spectral theorem states that any normal matrix A can be unitarily diagonalized: there exists a unitary matrix U such that U* A U is diagonal with eigenvalues on the diagonal. If A is Hermitian (self-adjoint), its eigenvalues are real; if A is unitary, its eigenvalues have modulus one. Thus A can be written as A = U Λ U*, where Λ is diagonal.

In a separable Hilbert space, the spectral theorem extends to normal operators via a projection-valued measure E on the spectrum σ(A). The operator A admits a spectral decomposition A = ∫ λ dE(λ). This yields a functional calculus: for any bounded Borel function f, f(A) = ∫ f(λ) dE(λ). For self-adjoint operators, the spectrum lies in the real line and the spectral measure provides a resolution of the identity. The theorem also allows joint spectral representations for commuting families of self-adjoint operators.

Applications and significance include diagonalization and simplification of operators, quantum mechanics, differential operators, signal processing, and numerical linear algebra. The spectral theorems require the underlying space to be a Hilbert space and typically assume normality or self-adjointness; without these conditions, a straightforward diagonalization may not be possible.