# spectral theorem

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.

 $MM^{\displaystyle\star}=M^{\displaystyle\star}M.$

## Spectral Theorem

Let $M:U\rightarrow U$ be a linear transformation of a unitary space. TFAE

1. 1.

The transformation $M$ is normal.

2. 2.

Letting

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

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. 3.

We can decompose $M$ as the sum

 $M=\sum_{\lambda\in\Lambda}\lambda P_{\lambda},$

where $\Lambda\in\mathbb{C}$ is a finite subset of complex numbers indexing a family of commuting orthogonal projections $P_{\lambda}:U\rightarrow U$, i.e.

 $P_{\lambda}{}^{\displaystyle\star}=P_{\lambda}\qquad P_{\lambda}P_{\mu}=\begin% {cases}P_{\lambda}&\lambda=\mu\\ 0&\lambda\neq\mu,\end{cases}$

and where WLOG

 $\sum_{\lambda\in\Lambda}P_{\lambda}=1_{U}.$
4. 4.

There exists an orthonormal basis of $U$ that diagonalizes $M$.

## Remarks.

1. 1.

Here are some important classes of normal operators, distinguished by the nature of their eigenvalues.

2. 2.

There is a well-known version of the spectral theorem for $\mathbb{R}$, 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. 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 SpectralTheorem 2013-03-22 12:45:49 2013-03-22 12:45:49 rmilson (146) rmilson (146) 9 rmilson (146) Theorem msc 15A23 msc 15A63 msc 15A18 DiagonalizableOperator normal operator