Representation of a normal linear operator (or diagonalizable matrix) as a sum or integral of projections onto invariant subspaces indexed by points of its spectrum, yielding a decomposition into eigencomponents in the discrete case or a projection-valued measure in the continuous case.

Principle

Exploit the spectral theorem: a normal (or self-adjoint) operator admits an orthogonal decomposition governed by its spectrum, enabling functional calculus by integrating scalar functions against spectral projections.

Demonstration

For a real symmetric matrix, compute an orthonormal eigenbasis and write the matrix as ∑ λ_i P_i where λ_i are eigenvalues and P_i are orthogonal projections onto the corresponding eigenspaces; for an unbounded self-adjoint operator use the spectral measure integral representation.

Misapplication

Applying spectral decomposition to a non-normal matrix or assuming distinct eigenvectors exist when the operator is defective; this leads to incorrect diagonalization claims and overlooks Jordan blocks or continuous spectrum components.

Consequence

Correct spectral decomposition simplifies analysis: it diagonalizes operators where possible, allows computation of operator functions (e.g., exponentials), clarifies stability and long-term behavior, and separates modes by spectral value.

Reversal

The opposite is the Jordan (or primary) decomposition emphasizing nilpotent and non-diagonalizable parts rather than orthogonal spectral projections; for non-normal operators one must use generalized eigenvectors and Jordan chains.

Boundary

Requires normality (or stronger hypotheses) in finite or infinite dimensions for the classical projection decomposition; many operators have continuous spectrum or are non-normal, in which case a pure orthogonal spectral decomposition either takes the form of a projection-valued integral or is not available.

Semantic Tension

Often contrasted with singular value decomposition: SVD applies to any matrix via orthonormal factorization into singular vectors and nonnegative singular values, while spectral decomposition depends on spectral properties (normality, self-adjointness) and yields eigenvalue-based projections.

Synthesis

Spectral decomposition is the formulation of an operator as a combination of spectral projections indexed by its spectrum, turning operator problems into tractable scalar problems via orthogonal eigenspaces in the discrete case or via projection-valued integrals for continuous spectra.