spectral resolution
There are two ways to write down a diagonalizable operator. One lists eigenvectors and eigenvalues. The other, the spectral resolution, repackages the same information using projections, which turns out to be far easier to compute with. If a normal operator T has distinct eigenvalues lambda_1, ..., lambda_k, the spectral resolution is T = lambda_1 P_1 + ... + lambda_k P_k, where each P_i is the orthogonal projection onto the eigenspace of lambda_i.
These projections obey three clean rules that make the decomposition a genuine resolution: each P_i is self-adjoint and idempotent (P_i^2 = P_i, an orthogonal projection); distinct ones are orthogonal to each other (P_i P_j = 0 when i != j); and together they reconstruct the identity, P_1 + ... + P_k = I. That last identity is called a resolution of the identity: the projections carve the whole space into perpendicular eigenspace pieces with no overlap and no gaps.
The beauty is that the projections do all the bookkeeping. Powers collapse instantly: T^2 = sum lambda_i^2 P_i because the cross terms P_i P_j vanish. Any polynomial p gives p(T) = sum p(lambda_i) P_i. This is the launchpad for the functional calculus, where p is replaced by any function f at all. Notice the grouping is by distinct eigenvalue, not by eigenvector — a repeated eigenvalue contributes one projection onto its whole (possibly multi-dimensional) eigenspace.
A two-eigenvalue operator as a weighted sum of complementary orthogonal projections.
The grouping is by distinct eigenvalue, so the number of projections k can be less than n. Each P_i projects onto a whole eigenspace at once, of dimension equal to that eigenvalue's multiplicity.