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In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as = for some scalar eigenvalue . [1] [2] [3] The solutions to this equation may ...
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.
Let (H, , ) be a real or complex Hilbert space and let A : H → H be a bounded, compact, self-adjoint operator.Then there is a sequence of non-zero real eigenvalues λ i, i = 1, …, N, with N equal to the rank of A, such that |λ i | is monotonically non-increasing and, if N = +∞, + =
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix, or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either (sometimes called the combinatorial Laplacian) or / / (sometimes called the normalized Laplacian), where is a diagonal matrix with ...
Let the same eigenvalue equation be solved using a basis set of dimension N + 1 that comprises the previous N functions plus an additional one. Let the resulting eigenvalues be ordered from the smallest, λ ′ 1, to the largest, λ ′ N+1. Then, the Rayleigh theorem for eigenvalues states that λ ′ i ≤ λ i for i = 1 to N.
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]
In the theory of ordinary differential equations, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the Sturm–Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in ordinary ...
In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system = that is perturbed from one with known eigenvectors and eigenvalues =. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues x 0 i , λ 0 i , i = 1 , … n {\displaystyle x_{0i},\lambda _{0i ...