Spectral gap

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In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system.

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In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes.

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*:

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In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.

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<span class="mw-page-title-main">Jeff Cheeger</span> American mathematician

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In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold M is a positive real number h(M) defined in terms of the minimal area of a hypersurface that divides M into two disjoint pieces. In 1970, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace–Beltrami operator on M to h(M). This proved to be a very influential idea in Riemannian geometry and global analysis and inspired an analogous theory for graphs.

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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 have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems.

In linear algebra, the eigengap of a linear operator is the difference between two successive eigenvalues, where eigenvalues are sorted in ascending order.

Superstrong approximation is a generalisation of strong approximation in algebraic groups G, to provide spectral gap results. The spectrum in question is that of the Laplacian matrix associated to a family of quotients of a discrete group Γ; and the gap is that between the first and second eigenvalues. Here Γ is a subgroup of the rational points of G, but need not be a lattice: it may be a so-called thin group. The "gap" in question is a lower bound for the difference of those eigenvalues.

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