spectral theoremSpectralTheoremForHermitianMatrices2002-06-07 21:08:242008-05-01 00:45:17Theoremnormal operator\usepackage{amsmath}
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\newcommand{\adj}{^{\displaystyle \star}}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.
$$MM\adj= M\adj M.$$
\paragraph{Spectral Theorem}
Let $M:U\rightarrow U$ be a linear transformation of a unitary space.
TFAE
\begin{enumerate}
\item The transformation $M$ is normal.
\item 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 $\lambda_1\neq
\lambda_2$.
\item 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.
$$
P_\lambda{}\adj = P_\lambda \qquad
P_\lambda P_\mu =
\begin{cases}
P_\lambda & \lambda = \mu \\
0 & \lambda\neq \mu,
\end{cases}
$$
and where WLOG
$$\sum_{\lambda\in\Lambda} P_\lambda = 1_U.$$
\item There exists an orthonormal basis of $U$ that diagonalizes $M$.
\end{enumerate}
\paragraph{Remarks.}
\begin{enumerate}
\item Here are some important classes of normal operators,
distinguished by the nature of their eigenvalues.
\begin{itemize}
\item Hermitian operators. Eigenvalues are real.
\item Unitary transformations. Eigenvalues lie on the unit circle,
i.e. the set of complex numbers of modulus 1.
\item Orthogonal projections. Eigenvalues are either 0 or 1.
\end{itemize}
\item 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.
\item 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.
\end{enumerate}