In mathematics, the Rayleigh theorem for eigenvalues pertains to the behavior of the solutions of an eigenvalue equation as the number of basis functions employed in its resolution increases. Rayleigh, Lord Rayleigh, and 3rd Baron Rayleigh are the titles of John William Strutt, after the death of his father, the 2nd Baron Rayleigh. Lord Rayleigh made contributions not just to both theoretical and experimental physics, but also to applied mathematics. The Rayleigh theorem for eigenvalues, as discussed below, enables the energy minimization that is required in many self-consistent calculations of electronic and related properties of materials, from atoms, molecules, and nanostructures to semiconductors, insulators, and metals. Except for metals, most of these other materials have an energy or a band gap, i.e., the difference between the lowest, unoccupied energy and the highest, occupied energy. For crystals, the energy spectrum is in bands and there is a band gap, if any, as opposed to energy gap. Given the diverse contributions of Lord Rayleigh, his name is associated with other theorems, including Parseval's theorem. For this reason, keeping the full name of "Rayleigh Theorem for Eigenvalues" avoids confusions.
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In mathematics, the Rayleightheoremforeigenvalues pertains to the behavior of the solutions of an eigenvalue equation as the number of basis functions...
Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms (such as Rayleigh quotient...
is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue problem. This yields an equation for the eigenvalues p ( λ )...
is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Given an...
_{\max }.} The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain...
problem for the Helmholtz equation, and so λ is known as a Dirichlet eigenvaluefor Ω. Dirichlet eigenvalues are contrasted with Neumann eigenvalues: eigenvalues...
the perturbation of a simple eigenvalue (see in multiplicity of eigenvalues). In the entry applications of eigenvalues and eigenvectors we find numerous...
possible) Eigenvalue algorithm — a numerical algorithm for locating the eigenvalues of a matrix Power iteration Inverse iteration Rayleigh quotient iteration...
contains 0. The spectral theorem shows that σ(T) consists of the eigenvalues {λn} of T and of 0 (if 0 is not already an eigenvalue). The set σ(T) is a compact...
x_{1}} is the corresponding eigenvector. Also (in the maximum theorem) subsequent eigenvalues λ k {\displaystyle \lambda _{k}} and eigenvectors x k {\displaystyle...
Moreover, this eigenvalue is the largest eigenvalue of M. Proof Assume the spectral theorem. Let the eigenvalues of M be λ 1 ≤ λ 2 ≤ ⋯ ≤ λ n {\displaystyle...
x\rangle |.} The numerical range is the range of the Rayleigh quotient. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact. W ( α A...
symmetric, the spectral theorem implies that A has n real-valued eigenvalues λ1 ≥ λ2 ≥ … ≥ λn. It is known that all these eigenvalues are in [−d, d] and more...
the Eckart–Young theorem [1936]. The singular values (in Σ) are the square roots of the eigenvalues of the matrix XTX. Each eigenvalue is proportional...
characterization of eigenvalues leads to the Rayleigh–Ritz method: choose an approximating u {\displaystyle u} as a linear combination of basis functions (for example...
shapes but identical eigenvalues (see figure on right). Cauchy–Kovalevskaya theorem In mathematics, the Cauchy–Kowalevski theorem (also written as the...
dimensions, the eigenvalues of D are real; the eigenspaces for distinct eigenvalues are orthogonal. It turns out that the eigenvalues can be described...
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