spectral theorem

Let $U$ be a finite-dimensional, 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

   $$\mathrm{\Lambda }=\{\lambda \in ℂ\mid M-\lambda E\text{ is singular}\},$$

   where $E$ is the identity operator, denote the spectrum (set of eigenvalues) of $M$, the corresponding eigenspaces

   $$E_{\lambda }=\ker (M-\lambda E),\lambda \in \mathrm{\Lambda }$$

   give an orthogonal, direct sum decomposition of $U$, i.e.

   $$U=\underset{\lambda \in \mathrm{\Lambda }}{\oplus }{E}_{\lambda },$$

   and $E_{{\lambda }_1}⟂E_{{\lambda }_2}$ 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 $\mathrm{\Lambda }\in ℂ$ is a finite subset of complex numbers indexing a family of commuting orthogonal projections $P_{\lambda }:U\to U$, i.e.

   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 well-known version of the spectral theorem for $ℝ$, 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.

Title	spectral theorem
Canonical name	SpectralTheorem
Date of creation	2013-03-22 12:45:49
Last modified on	2013-03-22 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