Compatible observables and complete bases

For Hermitian matrices A and B that commute, applying B to an A eigenvector keeps the result inside the same A eigenspace. When that eigenspace is one-dimensional, the vector is automatically also a B eigenvector. When it is degenerate, B can mix its vectors, but its Hermitian restriction can be diagonalized within that subspace. Repeating this operation constructs an orthonormal basis of simultaneous eigenvectors without disturbing the already diagonal first observable.

Degeneracy does not imply that every eigenvector of A is an eigenvector of B. It guarantees that a suitable basis can be chosen inside each degenerate eigenspace. If two listed joint eigenvalue tuples remain identical, the chosen commuting set has not resolved that remaining degeneracy. Complete here means enough joint labels to identify basis rays, not enough information to determine an arbitrary unknown superposition from one measurement on one copy.

Arrange the new orthonormal basis vectors as columns of a unitary matrix U in the old coordinates. The new coefficient vector is U dagger times the old one, and an observable matrix is conjugated by the same coordinate change. Transforming only the vector while leaving the observable matrix fixed usually describes an active change relative to a fixed apparatus. Confusing these operations can manufacture an apparent disagreement between two equivalent descriptions.

Unitarity preserves inner products and normalization, whereas mere invertibility does not. A general invertible coordinate system can still be used with its proper Gram matrix, but the simple sum of squared coefficients is then no longer automatically the physical norm. In the orthonormal bases used here, no such correction is needed. A matrix with determinant one can stretch some vectors and shrink others, so determinant preservation alone does not establish a valid unitary change between orthonormal quantum coordinates.

In finite dimension, commuting Hermitian matrices have commuting projectors and a common eigenbasis. Infinite-dimensional self-adjoint observables require a stronger spectral formulation: their spectral projections must commute for a joint sharp observable. A formal commutator vanishing on some common test functions does not automatically establish this strong commutation. Domains and continuous spectra prevent an unrestricted transfer of the finite-matrix proof to arbitrary differential operators.

Even when a compatible measurement exists, its implementation matters for disturbance inside degenerate eigenspaces. An ideal joint projective measurement of the refined labels is a specific instrument, not the only possible apparatus returning the same coarse A outcome. The joint probabilities follow from the commuting projectors, while statements about the state afterward need an update model. This separation becomes essential when generalized measurements and quantum instruments are introduced later in the module.