A _Complete set of compatible operators_ Are any complete set of [[Hermitian operators]] that have a common set of [eigenvalues and eigenvectors](Eigenvalues%20and%20eigenvectors.md) and each eigenvalue corresponds to a unique eigenvector. This allows us to then define an [orthonormal basis](Observable.md#Constructing%20observables) for a particular [Hilbert space](Hilbert%20Space.md) using their eigenvectors and eigenvectors. Any set of Hermitian operators that have a common eigenbasis and eigenvalues also [commute](Commutators%20in%20quantum%20mechanics.md) ([Proof](Complete%20sets%20of%20compatible%20operators.md#Proof%20that%20for%20any%20two%20commuting%20operators%20one%20may%20construct%20a%20common%20eigenbasis)). A single operator $A$ forms a trivial CSCO as long as it doesn't have a degenerate spectrum. %%See page 14 of McGreevy, this is used to study degenerate eigenspaces.%% %%what if one of the operators is non-Hermitian? like a for example if there's communitivity with a non-Hermitian unitary operator?%% --- # Proof that for any two commuting operators one may construct a common eigenbasis ## Non-degenerate case   #QuantumMechanics/QuantumMeasurement/QuantumObservables #QuantumMechanics/MathematicalFoundations #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/SpectralTheory