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 withits 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 correspondingeigenspaces

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

   give an orthogonal, direct sumdecomposition 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 numbersindexing 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 areal inner product spaces can diagonalized and that eigenvectorscorresponding to different eigenvalues are orthogonal. An even moredown-to-earth version of this theorem says that a symmetric, realmatrix can always be diagonalized by an orthonormal basis ofeigenvectors.

3. There are several versions of increasing sophistication of thespectral theorem that hold in infinite-dimensional, Hilbert spacesetting. In such a context one must distinguish between theso-called discrete and continuous (no corresponding eigenspace)spectrums, and replace the representing sum for the operator withsome kind of an integral. The definition of self-adjointness is alsoquite tricky for unbounded operators. Finally, there are versionsof the spectral theorem, of importance in theoretical quantummechanics, that can be applied to continuous 1-parameter groups ofcommuting, self-adjoint operators.