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17 relations: Compact operator on Hilbert space, Compression (functional analysis), Courant minimax principle, Eigenvalues and eigenvectors, Essential spectrum, Functional analysis, Hermitian matrix, Hilbert space, Limit point, Linear algebra, List of things named after Charles Hermite, Max–min inequality, Multiplicity (mathematics), Rayleigh quotient, Self-adjoint operator, Singular value, Spectrum (functional analysis).
Compact operator on Hilbert space
In functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.
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Compression (functional analysis)
In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator where P_K: H \rightarrow K is the orthogonal projection onto K. This is a natural way to obtain an operator on K from an operator on the whole Hilbert space.
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Courant minimax principle
In mathematics, the Courant minimax principle gives the eigenvalues of a real symmetric matrix.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Essential spectrum
In mathematics, the essential spectrum of a bounded operator is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible".
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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 (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
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Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and: Hermitian matrices can be understood as the complex extension of real symmetric matrices.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Limit point
In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.
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Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.
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List of things named after Charles Hermite
Numerous things are named after the French mathematician Charles Hermite (1822–1901).
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Max–min inequality
In mathematics, the max–min inequality is as follows: for any function f: Z × W → ℝ, \sup_ \inf_ f(z, w) \leq \inf_ \sup_ f(z, w).
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Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset.
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Rayleigh quotient
In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient R(M, x), is defined as: For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x^ to the usual transpose x'.
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Self-adjoint operator
In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle.
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Singular value
In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator acting between Hilbert spaces X and Y, are the square roots of the eigenvalues of the non-negative self-adjoint operator (where T* denotes the adjoint of T).
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Spectrum (functional analysis)
In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix.
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