Let $U$ be a finite-dimensional, unitary space and let $M:U\rightarrow U$ be an endomorphism. We say that $M$ is normal if it commutes with its Hermitian adjoint, i.e. $$MM\adj= M\adj M.$$

Spectral Theorem

1. The transformation $M$ is normal.
2. Letting $$\Lambda=\{\lambda\in\cnums \mid M-\lambda E \mbox{ is singular}\},$$ where $E$ is the identity operator, denote the spectrum (set of eigenvalues) of $M$ , the corresponding eigenspaces $$E_\lambda=\ker(M-\lambda E),\quad \lambda\in\Lambda$$ give an orthogonal, direct sum decomposition of $U$ , i.e. $$U=\bigoplus_{\lambda\in\Lambda} E_\lambda,$$ and $E_{\lambda_1} \perp E_{\lambda_2}$ for distinct eigenvalues $\lambda_1\neq \lambda_2$ .
3. We can decompose $M$ as the sum $$M = \sum_{\lambda\in \Lambda} \lambda P_\lambda,$$ where $\Lambda\in\cnums$ is a finite subset of complex numbers indexing a family of commuting orthogonal projections $P_\lambda:U\rightarrow U$ , i.e.
   
   and where WLOG $$\sum_{\lambda\in\Lambda} P_\lambda = 1_U.$$
4. There exists an orthonormal basis of $U$ that diagonalizes $M$ .

Remarks.

1. Here are some important classes of normal operators, distinguished by the nature of their eigenvalues.
   • Hermitian operators. Eigenvalues are real.
   • Unitary transformations. Eigenvalues lie on the unit circle, i.e. the set of complex numbers of modulus 1.
   • Orthogonal projections. Eigenvalues are either 0 or 1.
2. There is a well-known version of the spectral theorem for $\reals$ , namely that a self-adjoint (symmetric) transformation of a real inner product spaces can diagonalized and that eigenvectors corresponding to different eigenvalues are orthogonal. An even more down-to-earth version of this theorem says that a symmetric, real matrix can always be diagonalized by an orthonormal basis of eigenvectors.
3. There are several versions of increasing sophistication of the spectral theorem that hold in infinite-dimensional, Hilbert space setting. In such a context one must distinguish between the so-called discrete and continuous (no corresponding eigenspace) spectrums, and replace the representing sum for the operator with some kind of an integral. The definition of self-adjointness is also quite tricky for unbounded operators. Finally, there are versions of the spectral theorem, of importance in theoretical quantum mechanics, that can be applied to continuous 1-parameter groups of commuting, self-adjoint operators.