spectral decomposition
Spectral decomposition is the way of writing a Hermitian operator as a clean sum built from its own eigenvalues and eigenstates. Each term in the sum is an eigenvalue multiplied by a projector — an operator that singles out the part of any state lying along the corresponding eigenstate. The operator is thereby rebuilt entirely from its spectrum and its natural directions.
This decomposition is more than tidy bookkeeping; it lays bare exactly how an observable acts. Feeding in a state, each projector picks off that state's component along an eigenstate, and the matching eigenvalue scales it. The picture makes vivid why the possible measurement outcomes are the eigenvalues, and why the probability of each is the weight of the corresponding component, just as the Born rule prescribes.
Spectral decomposition also gives meaning to functions of an operator. To compute, say, the time-evolution operator built from the Hamiltonian, you simply apply the ordinary function to each eigenvalue while keeping the same projectors. This turns hard operator manipulations into easy arithmetic on numbers, and it underlies practical calculations throughout quantum mechanics.
A Hermitian operator is the sum of its eigenvalues weighting projectors onto its eigenstates.
Operators with continuous spectra, such as position, replace the discrete sum with an integral, but the spirit is identical. The spectral theorem is what guarantees a well-behaved Hermitian operator can always be decomposed this way.