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Singular value, the Glossary

Index Singular value

In mathematics, in particular functional analysis, the singular values of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator T^*T (where T^* denotes the adjoint of T).[1]

Table of Contents

  1. 29 relations: Banach space, Charles Royal Johnson, Compact operator, Condition number, Diagonal matrix, Eigenvalues and eigenvectors, Ellipsoid, Erhard Schmidt, Functional analysis, Hermitian adjoint, Hilbert space, Israel Gohberg, Ky Fan, Mark Krein, Mathematics, Matrix (mathematics), N-sphere, Normal matrix, Normed vector space, Operator norm, Poincaré separation theorem, Real number, Roger Horn, Schatten norm, Schur–Horn theorem, Singular value decomposition, Spectral theorem, Trace (linear algebra), Unitary matrix.

  2. Singular value decomposition

Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Charles Royal Johnson

Charles Royal Johnson (born January 28, 1948) is an American mathematician specializing in linear algebra.

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Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact closure in Y). Singular value and compact operator are operator theory.

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Condition number

In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument.

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Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices.

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Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector is a vector that has its direction unchanged by a given linear transformation. Singular value and Eigenvalues and eigenvectors are singular value decomposition.

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Ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

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Erhard Schmidt

Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century.

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Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.

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Hermitian adjoint

In mathematics, specifically in operator theory, each linear operator A on an inner product space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule where \langle \cdot,\cdot \rangle is the inner product on the vector space. Singular value and Hermitian adjoint are operator theory.

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Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Singular value and Hilbert space are operator theory.

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Israel Gohberg

Israel Gohberg (ישראל גוכברג; Изра́иль Цу́дикович Го́хберг; 23 August 1928 – 12 October 2009) was a Bessarabian-born Soviet and Israeli mathematician, most known for his work in operator theory and functional analysis, in particular linear operators and integral equations.

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Ky Fan

Ky Fan (樊𰋀,, September 19, 1914 – March 22, 2010) was a Chinese-born American mathematician.

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Mark Krein

Mark Grigorievich Krein (Марко́ Григо́рович Крейн, Марк Григо́рьевич Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Matrix (mathematics)

In mathematics, a matrix (matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.

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N-sphere

In mathematics, an -sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer.

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Normal matrix

In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose: The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras.

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Normed vector space

In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined.

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Operator norm

In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its. Singular value and operator norm are operator theory.

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Poincaré separation theorem

In mathematics, the Poincaré separation theorem, also known as the Cauchy interlacing theorem, gives some upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B.

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Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

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Roger Horn

Roger Alan Horn (born January 19, 1942) is an American mathematician specializing in matrix analysis.

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Schatten norm

In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm. Singular value and Schatten norm are operator theory.

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Schur–Horn theorem

In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues.

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Singular value decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation. Singular value and singular value decomposition are operator theory.

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Spectral theorem

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). Singular value and spectral theorem are singular value decomposition.

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Trace (linear algebra)

In linear algebra, the trace of a square matrix, denoted, is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of.

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Unitary matrix

In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose, that is, if U^* U.

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See also

Singular value decomposition

References

[1] https://en.wikipedia.org/wiki/Singular_value

Also known as Singular Values.