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Its eigenfunctions form a basis of the function space on which the operator is defined [5] As a consequence, in many important cases, the eigenfunctions of the Hermitian operator form an orthonormal basis. In these cases, an arbitrary function can be expressed as a linear combination of the eigenfunctions of the Hermitian operator.
The definitions of eigenvalue and eigenvectors of a linear transformation T remains valid even if the underlying vector space is an infinite-dimensional Hilbert or Banach space. A widely used class of linear transformations acting on infinite-dimensional spaces are the differential operators on function spaces.
These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables, as well as to find eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid, which is presented as a Kronecker sum of discrete Laplacians in one-dimension.
The eigenfunctions of the position operator (on the space of tempered distributions), represented in position space, are Dirac delta functions. Informal proof. To show that possible eigenvectors of the position operator should necessarily be Dirac delta distributions, suppose that ψ {\displaystyle \psi } is an eigenstate of the position ...
The compatibility theorem tells us that a common basis of eigenfunctions of ^ and ^ can be found. Now if each pair of the eigenvalues ( a n , b n ) {\displaystyle (a_{n},b_{n})} uniquely specifies a state vector of this basis, we claim to have formed a CSCO: the set { A , B } {\displaystyle \{A,B\}} .
If we use the third choice of domain (with periodic boundary conditions), we can find an orthonormal basis of eigenvectors for A, the functions ():=. Thus, in this case finding a domain such that A is self-adjoint is a compromise: the domain has to be small enough so that A is symmetric, but large enough so that D ( A ∗ ) = D ( A ...
The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The eigenvalues of the Hessian at that point are the principal curvatures of the function, and the eigenvectors are the principal directions of curvature.
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. [1]