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Jacobi Matrix (operator)
A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel measure. This operator is named after Carl Gustav Jacob Jacobi. The name derives from a theorem from Jacobi, dating to 1848, stating that every symmetric matrix over a principal ideal domain is congruent to a tridiagonal matrix. Self-adjoint Jacobi operators The most important case is the one of self-adjoint Jacobi operators acting on the Hilbert space of square summable sequences over the positive integers \ell^2(\mathbb). In this case it is given by :Jf_0 = a_0 f_1 + b_0 f_0, \quad Jf_n = a_n f_ + b_n f_n + a_ f_, \quad n>0, where the coefficients are assumed to satisfy :a_n >0, \quad b_n \in \mathbb. The operator will be bounded if and only if the coefficients are bounded. There are close connections with the theory of orthogonal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Toda Lattice
The Toda lattice, introduced by , is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system. It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian :\begin H(p,q) &= \sum_ \left(\frac +V(q(n+1,t)-q(n,t))\right) \end and the equations of motion :\begin \frac p(n,t) &= -\frac = e^ - e^, \\ \frac q(n,t) &= \frac = p(n,t), \end where q(n,t) is the displacement of the n-th particle from its equilibrium position, and p(n,t) is its momentum (mass m=1), and the Toda potential V(r)=e^+r-1. Soliton solutions Soliton solutions are solitary waves spreading in time with no change to their shape and size and interacting with each other in a particle-like way. The general N-soliton solution of the equation is : \begin q_N(n,t)=q_+ + \log \frac , \end where :C_N(n,t)=\Bigg(\frac\Bigg)_, with :\gamma_j(n,t)=\gamma_j\,e^ where \kap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hankel Matrix
In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example, \qquad\begin a & b & c & d & e \\ b & c & d & e & f \\ c & d & e & f & g \\ d & e & f & g & h \\ e & f & g & h & i \\ \end. More generally, a Hankel matrix is any n \times n matrix A of the form A = \begin a_0 & a_1 & a_2 & \ldots & a_ \\ a_1 & a_2 & & &\vdots \\ a_2 & & & & a_ \\ \vdots & & & a_ & a_ \\ a_ & \ldots & a_ & a_ & a_ \end. In terms of the components, if the i,j element of A is denoted with A_, and assuming i \le j, then we have A_ = A_ for all k = 0,...,j-i. Properties * Any Hankel matrix is symmetric. * Let J_n be the n \times n exchange matrix. If H is an m \times n Hankel matrix, then H = T J_n where T is an m \times n Toeplitz matrix. ** If T is real symmetric, then H = T J_n will have the same eigenvalues as T up to sign. * The Hilbert matrix is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the correspondi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Shift Operator
In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In time series analysis, the shift operator is called the '' lag operator''. Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. The notion of triangulated category is a categorified analogue of the shift operator. Definition Functions of a real variable The shift operator (where ) takes a fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hessenberg Matrix
In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal. They are named after Karl Hessenberg. A Hessenberg decomposition is a matrix decomposition of a matrix A into a unitary matrix P and a Hessenberg matrix H such that PHP^*=A where P^* denotes the conjugate transpose. Definitions Upper Hessenberg matrix A square n \times n matrix A is said to be in upper Hessenberg form or to be an upper Hessenberg matrix if a_=0 for all i,j with i > j+1. An upper Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero, i.e. if a_ \neq 0 for all i \in \. Lower Hessenberg matrix A square n \times n matrix A is said to be in lower Hessenberg form or to be a lower Hessenberg matrix if its transpose is an upper Hessenberg matrix or equivalently if a_=0 for al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hessenberg Operator
Hessenberg may refer to: People: *Gerhard Hessenberg (1874–1925), German mathematician * Karl Hessenberg (1904–1959), German mathematician and engineer *Kurt Hessenberg (1908–1994), German composer and professor at the Hochschule für Musik und Darstellende Kunst in Frankfurt am Main Mathematics: *Hessenberg matrix In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above ..., one that is "almost" triangular * Hessenberg variety, a family of subvarieties of the full flag variety which are defined by a Hessenberg function h and a linear transformation X {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bergman Polynomial
Bergman is a surname of German, Swedish, Dutch and Yiddish origin meaning 'mountain man', or sometimes (only in German) 'miner'.https://www.ancestry.com/name-origin?surname=bergmann People * Abraham Bergman (1932–2023), American pediatrician *Alan Bergman (born 1925), American songwriter *Alan Bergman (1943–2010), American ballet dancer * Alfred Bergman (1889–1961), American baseball and football player *Amanda Bergman (born 1987), Swedish musician *Andrew Bergman (born 1945), American film director * Anita Bergman, Canadian politician *Annie Bergman (1889-1987), Swedish artist, writer, and children’s book author. *Bo Bergman (1869–1967), Swedish poet * Borah Bergman (1926–2012), American pianist * Cam Bergman (born 1983), Canadian lacrosse player *Carl Bergman (born 1987), Swedish tennis player *Carl Johan Bergman (born 1978), Swedish biathlete *Charlotte Bergman (1903–2002), Belgian art collector and philanthropist *Christian Bergman (born 1988), American baseball ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Holomorphic Functions
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is '' analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term ''analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Square-integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty, +\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as ,b/math> for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the L^p space with p = 2. Among the L^p spaces, the class of square integrable functions is unique in being compatible with an inner product, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
