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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] |
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] |