orthogonal projections from spectrum
The spectral resolution assigns to each eigenvalue an orthogonal projection P_i. It is worth looking at these projections as objects in their own right, because they are the finite-dimensional seed of one of the deepest ideas in operator theory: the projection-valued measure. The projection P_i is the orthogonal projection onto the eigenspace of lambda_i; it answers the question how much of any vector lives in that eigenspace.
Think of the spectrum as a tiny measure space. To a single eigenvalue you assign its projection P_i; to a set of eigenvalues you assign the sum of the corresponding projections. This assignment behaves exactly like a measure, but the values are projections instead of numbers: it is additive on disjoint sets (P for a union is the sum of the parts, since the projections are mutually orthogonal), the empty set maps to 0, and the whole spectrum maps to the identity I (the resolution of the identity). The operator itself is recovered by integrating the identity function against this measure: T = sum lambda_i P_i.
In finite dimensions this is just bookkeeping for the spectral resolution, but it generalizes spectacularly. For a self-adjoint operator on an infinite-dimensional Hilbert space the spectrum can be a continuum with no eigenvectors at all, and the sum T = sum lambda_i P_i becomes an integral T = integral lambda dP(lambda) against a genuine projection-valued measure. That integral form is the true spectral theorem of functional analysis and the rigorous foundation of quantum observables. Everything starts from these finite-dimensional spectral projections.
Spectral projections assemble into a measure whose values are projections, summing to the identity.
P_i can be written explicitly as q_i q_i^* for a one-dimensional eigenspace, or as a sum of such rank-one projections over an orthonormal basis of a larger eigenspace. It is self-adjoint and idempotent: P_i = P_i^* = P_i^2.