What is spectral theorem in linear algebra?
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). In more abstract language, the spectral theorem is a statement about commutative C*-algebras.
What does the spectral theorem say?
The spectral theorem shows that there is no loss of generality in assuming that A is the multiplication induced by X, say, on a measure space X with measure ,u.
Why is the spectral theorem important?
The spectral theorem in the finite-dimensional case is important in spectral graph theory: the adjacency matrix and Laplacian of an undirected graph are both symmetric, hence both have real eigenvalues and an orthonormal basis of eigenvectors, and this is important to many applications of these matrices, e.g. to the …
Why is it called the spectral theorem?
Since the theory is about eigenvalues of linear operators, and Heisenberg and other physicists related the spectral lines seen with prisms or gratings to eigenvalues of certain linear operators in quantum mechanics, it seems logical to explain the name as inspired by relevance of the theory in atomic physics.
What is the spectral theorem in quantum mechanics?
The spectral theorem, a major result in functional analysis, states that any normal (A linear operator is normal if it is closed, densely defined, and it commutes with its adjoint.) operator on a Hilbert space H can be diagonalized, i.e. it is unitarily equivalent to a multiplication operator [1].
What is spectral decomposition in linear algebra?
When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called “spectral decomposition”, derived from the Spectral theorem. …
What is seismic spectral decomposition?
Time-frequency analysis or spectral decomposition is a technique that allows geophysicists to visualize frequency content of seismic data along a time axis. Over the last decades, numerous techniques of time-frequency analysis and case studies have been published in literature.
Why is it called spectral theory?
What is spectral representation?
The spectral decomposition is thus an analogue for stationary stochastic processes of the more familiar Fourier representation of deterministic functions. The analysis of stationary processes by means of their spectral representations is often referred to as the “frequency domain” analysis of time series.
What is spectral math?
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.
What is a spectrum in math?
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.