A decomposition of an operator into components associated with its spectrum, typically expressed as a sum of scalar eigenvalues times projection operators onto the corresponding eigenspaces; for normal operators on inner-product spaces these projections are orthogonal.

Principle

When an operator is diagonalizable, it admits a representation A = Σ λ P_λ where {P_λ} are pairwise orthogonal idempotent projections summing to the identity and λ ranges over the spectrum; in infinite-dimensional or nonnormal contexts the spectral theorem generalizes this via spectral measures.

Demonstration

A real symmetric matrix A can be written as A = Σ_i λ_i u_i u_i^T where the u_i form an orthonormal eigenbasis and u_i u_i^T are orthogonal rank-one projections; this decomposition yields an immediate formula for functions of A like exp(A) or A^k.

Misapplication

Applying spectral decomposition with orthogonal projections to nonnormal matrices without diagonalizability; treating nonorthogonal generalized eigenspaces as if they were spectral projections leads to incorrect functional calculations and stability claims.

Consequence

Spectral decomposition furnishes a functional calculus (apply scalar functions to eigenvalues and recombine via projections), simplifies analysis of dynamics, and underpins many algorithms in numerical linear algebra and quantum theory when the required hypotheses hold.

Reversal

The reversal is the Jordan perspective: instead of orthogonal spectral projections one accepts generalized eigenspaces and nilpotent parts, expressing the operator as a sum of Jordan blocks rather than a direct sum of eigenspace projections.

Boundary

Full spectral decomposition with orthogonal projections requires diagonalizability or normality in finite dimensions (or a self-adjointness / normality hypothesis in infinite dimensions); otherwise one must use Jordan form, Schur decomposition, or the continuous spectral measure framework.

Semantic Tension

Tension between spectral decomposition and singular value decomposition (SVD): spectral decomposition diagonalizes normal operators with respect to eigenvectors, while SVD applies to arbitrary matrices producing orthonormal singular vectors and nonnegative singular values not equal to eigenvalues in general.

Synthesis

Spectral decomposition expresses an operator as a recombination of spectral projections weighted by eigenvalues, giving a canonical functional representation when eigenvectors (or spectral measures) are available and separating pure spectral action from nilpotent distortions handled by other canonical forms.