The spectral theorem for self-adjoint operators

For a finite self-adjoint matrix A, each distinct eigenvalue λ has an orthogonal projection P_λ onto its eigenspace. These projections add to the identity, are pairwise orthogonal, and reconstruct A as a sum of λP_λ. Degenerate eigenvalues require one projection onto the whole eigenspace, not an arbitrary preferred eigenvector. The operator is unchanged when an orthonormal basis within a degenerate eigenspace is rotated. Spectral projections capture the geometry without that coordinate ambiguity.

A projection-valued measure E assigns an orthogonal projection E(B) to each Borel subset B of the real line. Disjoint sets correspond to orthogonal projections, and countable additivity is interpreted in the strong operator sense. For a bounded self-adjoint operator, A is the spectral integral of λ against E. Multiplication by x on L2(0,1) has E(B)f equal to the indicator of B times f. Its projections are meaningful even though it has no nonzero eigenfunctions.

For a bounded Borel function g, the functional calculus defines g(A) by integrating g against the spectral measure. In finite dimensions this simply replaces each eigenvalue λ by g(λ) while preserving the spectral projections. One can define a positive square root, a resolvent, or a time-evolution operator without expanding an arbitrary matrix power by hand. If g is continuous on the compact spectrum of a bounded self-adjoint A, its operator norm is the maximum of |g| on that spectrum.

Scalar expressions still have domains. The inverse A^{-1} is a bounded everywhere-defined operator only when zero is outside the spectrum. A square root as a positive self-adjoint operator requires A to be nonnegative. For an unbounded self-adjoint A, an unbounded function g produces an operator defined only on vectors with a finite weighted spectral integral. Writing g(A) does not make that integrability condition disappear, and composing unbounded functions requires compatible domains.

For a compact self-adjoint operator, the Hilbert space is the orthogonal direct sum of its kernel and the closure of the span of its nonzero eigenvectors. Its nonzero eigenvalues are real and can only accumulate at zero. This is an eigenbasis representation after including a basis of the kernel. General bounded self-adjoint operators need not be compact, and their spectral measures may contain a continuous part. Importing a discrete sum into that situation can erase the main feature of the spectrum.

Spectral measures also organize observations. For a normalized x, μ_x is a probability measure over spectral values. In a finite state it assigns weights equal to squared projection lengths. This is a mathematical probability distribution associated with a particular vector and observable. Its use in quantum theory adds physical interpretation, but the spectral theorem itself does not simulate measurements or choose an outcome. The finite-mode lab below illustrates a representation choice and truncation, not a construction of arbitrary spectral measures.