Spectraldecompositionisafundamentalconceptinlinearalgebraandmatrixtheory.Itreferstotheprocessofbreakingdownasquarematrixintoitsconstituenteigenvaluesandeigenvectors.ForadiagonalizablematrixA,spectraldecompositionexpressesAasasumofsimplermatricesscaledbyitseigenvalues.Specifically,ifAhaseigenvaluesλ₁,λ₂,...,λₙwithcorrespondingeigenvectorsv₁,v₂,...,vₙ,thenAcanbewrittenasA=PDP⁻¹,wherePisthematrixofeigenvectorsandDisthediagonalmatrixofeigenvalues.Anequivalentformulationistheouterproductrepresentation:A=Σλᵢv�vᵢᵀ,summingoveralleigenvaluesandeigenvectors.Thisdecompositionrevealstheunderlyingstructureofthematrixandisparticularlyusefulforsymmetricmatrices,whichalwayshaverealeigenvaluesandorthogonaleigenvectors.Spectraldecompositionhaswideapplicationsacrossmathematicsandengineering,includingprincipalcomponentanalysis(PCA),solvingsystemsofdifferentialequations,quantummechanics,andvibrationanalysis.Itprovidesinsightsintomatrixpropertiesandsimplifiescomputationsinvolvingmatrixpowersorexponentials.
