Let be a linear transformation of a unitary space. TFAE
The transformation is normal.
Here are some important classes of normal operators, distinguished by the nature of their eigenvalues.
Hermitian operators. Eigenvalues are real.
Orthogonal projections. Eigenvalues are either 0 or 1.
There is a well-known version of the spectral theorem for , 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.
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.
|Date of creation||2013-03-22 12:45:49|
|Last modified on||2013-03-22 12:45:49|
|Last modified by||rmilson (146)|