spectral theorem
Let $U$ be a finitedimensional, unitary space and let $M:U\to U$ be an endomorphism^{}. We say that $M$ is normal if it commutes with its Hermitian adjoint, i.e.
$$M{M}^{\star}={M}^{\star}M.$$ 
Spectral Theorem
Let $M:U\to U$ be a linear transformation of a unitary space. TFAE

1.
The transformation^{} $M$ is normal.

2.
Letting
$$\mathrm{\Lambda}=\{\lambda \in \u2102\mid M\lambda E\text{is singular}\},$$ where $E$ is the identity operator^{}, denote the spectrum^{} (set of eigenvalues^{}) of $M$, the corresponding eigenspaces^{}
$${E}_{\lambda}=\mathrm{ker}(M\lambda E),\lambda \in \mathrm{\Lambda}$$ give an orthogonal^{}, direct sum^{} decomposition of $U$, i.e.
$$U=\underset{\lambda \in \mathrm{\Lambda}}{\oplus}{E}_{\lambda},$$ and ${E}_{{\lambda}_{1}}\u27c2{E}_{{\lambda}_{2}}$ for distinct eigenvalues ${\lambda}_{1}\ne {\lambda}_{2}$.

3.
We can decompose $M$ as the sum
$$M=\sum _{\lambda \in \mathrm{\Lambda}}\lambda {P}_{\lambda},$$ where $\mathrm{\Lambda}\in \u2102$ is a finite subset of complex numbers indexing a family of commuting orthogonal projections^{} ${P}_{\lambda}:U\to U$, i.e.
$${P}_{\lambda}{}^{\star}={P}_{\lambda}\mathit{\hspace{1em}\hspace{1em}}{P}_{\lambda}{P}_{\mu}=\{\begin{array}{cc}{P}_{\lambda}\hfill & \lambda =\mu \hfill \\ 0\hfill & \lambda \ne \mu ,\hfill \end{array}$$ and where WLOG
$$\sum _{\lambda \in \mathrm{\Lambda}}{P}_{\lambda}={1}_{U}.$$ 
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 wellknown version of the spectral theorem^{} for $\mathbb{R}$, namely that a selfadjoint (symmetric^{}) transformation of a real inner product spaces^{} can diagonalized and that eigenvectors^{} corresponding to different eigenvalues are orthogonal. An even more downtoearth 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 infinitedimensional, Hilbert space^{} setting. In such a context one must distinguish between the socalled discrete and continuous^{} (no corresponding eigenspace) spectrums, and replace the representing sum for the operator^{} with some kind of an integral. The definition of selfadjointness 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 1parameter groups of commuting, selfadjoint operators.
Title  spectral theorem 

Canonical name  SpectralTheorem 
Date of creation  20130322 12:45:49 
Last modified on  20130322 12:45:49 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  9 
Author  rmilson (146) 
Entry type  Theorem 
Classification  msc 15A23 
Classification  msc 15A63 
Classification  msc 15A18 
Related topic  DiagonalizableOperator 
Defines  normal operator 