## [Spectral Decomposition](/spectral-decomposition) ## [Spectral Decomposition](https://quantummech.quantumdictionary.io/spectral-decomposition-0) [![Quantum Mechanics Dictionary](/sites/default/files/styles/large/public/2026-01/Quantum%20Mechanics.png.webp?itok=tC_FGCW0)](/topic-specific-dictionaries/modern-physics/quantum-mechanics) - Modern Physics - **Quantum Mechanics Dictionary** Definition A quantum mechanics concept defining a model element, mathematical object, or experimental method used to predict measurable outcomes. It applies when required assumptions and definitions are specified and yields computable probabilities and expectation values. It does not ensure correctness without validation of approximations, numerical stability, and consistency of units and conventions. ## [Spectral Decomposition](https://mathlogic.quantumdictionary.io/spectral-decomposition-1) [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic) - Natural & Formal Sciences - **Mathematics & Logic Dictionary** Definition 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. ## [Spectral Decomposition](https://algebra.quantumdictionary.io/spectral-decomposition-2) [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra) - Pure Mathematics - **Algebra Dictionary** Definition 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.