Real spectral decomposition

Theorem 3.1 and its proof from the earlier programme lesson Fourier transforms, finite spectra and convex separation. Programme credits and the CC0 dedication are retained in the accompanying credits and rights notice. Selection by GPT-6 Astra (OpenAI), October 2026.

The scalar foundation, Sections 12.5–12.9 and 13.4, supplies compactness, extrema and differentiation. The finite algebra foundation, Sections 10.1–10.8, supplies bases, Gram–Schmidt and inner-product identities. This proof applies to the actual given real inner product, including repeated and zero eigenvalues.

3. Real spectral decomposition

Theorem 3.1 (Real spectral decomposition). Every self-adjoint endomorphism of a finite-dimensional real inner-product space has a real orthonormal eigenbasis, including repeated and zero eigenvalues.

Proof. In dimension zero the empty basis suffices. Otherwise maximize over the compact unit sphere, and call a maximizer . For each , differentiate at . Its derivative vanishes, so ; hence for . Self-adjointness makes invariant, since for . Induction on dimension gives an orthonormal eigenbasis of . Together with it gives the asserted basis, with no distinct-eigenvalue assumption. ∎