spectral projection
When a matrix is diagonalizable, the space splits cleanly into eigenspaces, and any vector can be decomposed into one piece per eigenspace. A spectral projection P_i is the operator that performs that decomposition: it extracts the component of a vector lying in the eigenspace E_lambda_i, while discarding the components living in all the other eigenspaces.
Formally, if A is diagonalizable with distinct eigenvalues lambda_1, ..., lambda_m, there are unique projections P_1, ..., P_m satisfying three properties: each P_i is idempotent (P_i^2 = P_i), they are mutually orthogonal as projections (P_i P_j = 0 for i != j), and they sum to the identity (P_1 + ... + P_m = I). Each P_i projects onto E_lambda_i along the sum of the other eigenspaces.
The payoff is the spectral decomposition A = lambda_1 P_1 + ... + lambda_m P_m. This rewrites A as a weighted sum of projections, and it makes functions of A trivial: A^k = sum lambda_i^k P_i, e^A = sum e^lambda_i P_i, and in general f(A) = sum f(lambda_i) P_i. For symmetric or Hermitian matrices, these projections are orthogonal projections, which is the heart of the spectral theorem.
The spectral decomposition: A is a sum of its eigenvalues weighting their eigenprojections.
Spectral projections are orthogonal (self-adjoint) only when A is normal (e.g. symmetric/Hermitian). For a general diagonalizable A they are oblique projections — still idempotent and summing to I, but not perpendicular.