Search results
Results from the WOW.Com Content Network
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. [1] It is a result of studies of linear algebra and the solutions of systems of linear equations and their ...
Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry ...
This conclusion is also true for a more general class of operators. A unitary operator is normal. By the spectral theorem, a bounded operator on a Hilbert space H is normal if and only if it is equivalent (after identification of H with an space) to a multiplication operator. It can be shown that the approximate point spectrum of a bounded ...
Spectral methods and finite-element methods are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains (compact support).
In mathematics, the spectrum of a matrix is the multiset of the eigenvalues of the matrix. In functional analysis, the concept of the spectrum of a bounded operator is a generalization of the eigenvalue concept for matrices. In algebraic topology, a spectrum is an object representing a generalized cohomology theory.
In mathematics, the spectrum of a matrix is the set of its eigenvalues. [ 1 ] [ 2 ] [ 3 ] More generally, if T : V → V {\displaystyle T\colon V\to V} is a linear operator on any finite-dimensional vector space , its spectrum is the set of scalars λ {\displaystyle \lambda } such that T − λ I {\displaystyle T-\lambda I} is not invertible .
The stable homotopy category, or homotopy category of (CW) spectra is defined to be the category whose objects are spectra and whose morphisms are homotopy classes of maps between spectra. Many other definitions of spectrum, some appearing very different, lead to equivalent stable homotopy categories.
In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank of the corresponding Riesz projector is finite.