spectral theorem (normal)

Self-adjoint operators are not the only ones with an orthonormal eigenbasis. Over the complex numbers there is a single clean condition that captures every such operator at once. An operator N is called normal when it commutes with its own adjoint: N N^* = N^* N. Self-adjoint operators (N = N^*), unitary operators (N^* = N^-1), and skew-adjoint operators (N^* = -N) are all special cases, but normality is the broadest umbrella.

The complex spectral theorem states: an operator on a finite-dimensional complex inner product space is unitarily diagonalizable if and only if it is normal. Unitarily diagonalizable means there is an orthonormal basis of eigenvectors, equivalently N = Q D Q^* with Q unitary and D diagonal. This is the most general finite-dimensional spectral statement: it draws the exact boundary of which operators enjoy a perpendicular eigenbasis. Everything inside the boundary is normal; everything normal is inside.

The eigenvalues here can be any complex numbers, which is why the theorem lives over C. Each special class is just a constraint on where those eigenvalues sit: self-adjoint forces them onto the real line, unitary forces them onto the unit circle, skew-adjoint onto the imaginary axis. The proof typically goes through Schur's theorem (every complex matrix is unitarily triangularizable), then shows that for a normal matrix the triangular factor must actually be diagonal, because the commuting condition kills every off-diagonal entry.